// Numbas version: finer_feedback_settings
{"showQuestionGroupNames": false, "question_groups": [{"name": "Group", "pickQuestions": 1, "pickingStrategy": "all-ordered", "questions": [{"name": "True/false statements about properties of continuity and limits, ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"v": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "name": "v"}, "f1": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"<p>If for some sequence {$u_n$} converging to $c$, the sequence {$f(u_n)$} converges to $l$, then $f(x)&nbsp;\\\\to&nbsp;l$ as $x&nbsp;\\\\to&nbsp;c$.</p>\"", "description": "", "name": "f1"}, "ch2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(u=1,tr4,if(u=2,tr5,tr6))", "description": "", "name": "ch2"}, "u": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "name": "u"}, "f4": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"<p>If $f(x)&nbsp;\\\\not\\\\to&nbsp;l$ as $x$ tends to $c$, then $f(x_n)&nbsp;\\\\not\\\\to&nbsp;l$ as $n&nbsp;\\\\to \\\\infty$&nbsp;for every sequence {$x_n$} converging to $c$.</p>\"", "description": "", "name": "f4"}, "f5": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"<p>If the limit of $f(x)$ as $x&nbsp;\\\\to&nbsp;c$ exists then the limit is $f(c)$.</p>\"", "description": "", "name": "f5"}, "h": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "description": "", "name": "h"}, "f2": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"<p>If for some sequence {$x_n$} converging to $c$, the sequence {$f(x_n)$} converges to $f(c)$, then the function $f$ is continuous at $c$.</p>\"", "description": "", "name": "f2"}, "f3": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"<p>If $f(x)&nbsp;\\\\to&nbsp;\\\\ell$ as $x&nbsp;\\\\to&nbsp;c$ and if $g(x)&nbsp;\\\\to&nbsp;m$ as $x&nbsp;\\\\to&nbsp;c$, then $\\\\dfrac{f(x)}{g(x)}&nbsp;\\\\to \\\\dfrac{\\\\ell}{m}$ as $x&nbsp;\\\\to&nbsp;c$.</p>\"", "description": "", "name": "f3"}, "tr5": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"<p>There exists a function $f$ such that the limit of $f(x)$ as $x&nbsp;\\\\to&nbsp;c$ exists and $f(c)$ exists, but $f$ is not continuous at $c$.</p>\"", "description": "", "name": "tr5"}, "ch1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(t=1,tr1,if(t=2,tr2,tr3))", "description": "", "name": "ch1"}, "tr6": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"<p>If the limit of $f(x)$ as $x&nbsp;\\\\to&nbsp;c$ exists and the limit is $f(c)$, then $f$ is continuous at $c$.</p>\"", "description": "", "name": "tr6"}, "tr2": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"<p>If a function $f$ is continuous at $c$, then for any sequence {$x_n$} converging to $c$, the sequence {$f(x_n)$} converges to $f(c)$.</p>\"", "description": "", "name": "tr2"}, "f": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "name": "f"}, "tr3": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"<p>If for every sequence {$x_n$} converging to $c$, the sequence {$f(x_n)$} converges to $f(c)$, then the function $f$ is continuous at $c$.</p>\"", "description": "", "name": "tr3"}, "f6": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"<p>If the limit of $f(x)$ as $x&nbsp;\\\\to&nbsp;c$ exists and if $f(c)$ exists, then $f$ is continuous at $c$.</p>\"", "description": "", "name": "f6"}, "ch3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(v=1,f1,if(v=2,f2,f3))", "description": "", "name": "ch3"}, "ch4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(f=1,f4,if(f=2,f5,f6))", "description": "", "name": "ch4"}, "tr4": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"<p>If $f(x)&nbsp;\\\\not\\\\to \\\\ell$ as $x&nbsp;\\\\to&nbsp;c$, then $f(x_n)&nbsp;\\\\not\\\\to \\\\ell$ as $n \\\\to \\\\infty$ for some sequence {$x_n$} converging to $c$.</p>\"", "description": "", "name": "tr4"}, "tr1": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"<p>If $f(x)&nbsp;\\\\to \\\\ell$ as $x&nbsp;\\\\to&nbsp;c$ and if $x_n&nbsp;\\\\to&nbsp;c$ as $n \\\\to \\\\infty$ (with each $x_n&nbsp;\\\\neq&nbsp;c$), then $f(x_n)&nbsp;\\\\to \\\\ell$ as $n \\\\to \\\\infty$.</p>\"", "description": "", "name": "tr1"}, "g": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "name": "g"}, "t": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "name": "t"}}, "ungrouped_variables": ["f1", "f2", "f3", "f4", "f5", "f6", "t", "tr1", "u", "tr2", "tr3", "tr4", "tr5", "tr6", "g", "f", "h", "ch1", "ch2", "ch3", "ch4", "v"], "functions": {}, "parts": [{"showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "\n  \n  \n  <p>[[0]]</p>\n  \n  \n  \n", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"displayType": "checkbox", "showCellAnswerState": true, "type": "m_n_x", "maxAnswers": 0, "shuffleChoices": true, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "minAnswers": 0, "minMarks": 0, "shuffleAnswers": false, "variableReplacements": [], "layout": {"expression": ""}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "matrix": [[1, -1], [1, -1], ["-1", "1"], [-1, 1]], "choices": ["<p>{Ch1}</p>", "<p>{Ch2}</p>", "<p>{Ch3}</p>", "<p>{Ch4}</p>"], "unitTests": [], "marks": 0, "scripts": {}, "maxMarks": 0, "showCorrectAnswer": true, "answers": ["True", "False"], "warningType": "none"}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "sortAnswers": false}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>Answer the following question on continuity and limits of&nbsp;functions. You may assume that the functions $f$ are $:\\mathbb{R} \\to \\mathbb{R}$. Note that every correct answer is worth 1 mark, but every wrong answer loses a mark.</p>", "tags": ["checked2015", "continuous", "convergence", "convergent sequences", "limits", "sequence", "sequences"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Multiple response question (2 correct out of 4) covering properties of continuity and limits of functions. Selection of questions from a pool.</p>"}, "advice": "<p>You should be able to work out the correct answers from your notes.</p>"}, {"name": "True/false statements about continuity and differentiability, ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "parts": [{"customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "\n  \n  \n  <p>[[0]]</p>\n  \n  \n  \n", "unitTests": [], "sortAnswers": false, "scripts": {}, "gaps": [{"displayType": "checkbox", "type": "m_n_x", "maxAnswers": 0, "shuffleChoices": true, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "minAnswers": 0, "minMarks": 0, "shuffleAnswers": false, "variableReplacements": [], "layout": {"expression": ""}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "matrix": [[1, -1], [1, -1], ["-1", "1"], [-1, 1]], "choices": ["<p>{Ch1}</p>", "<p>{Ch2}</p>", "<p>{Ch3}</p>", "<p>{Ch4}</p>"], "unitTests": [], "answers": ["True", "False"], "scripts": {}, "maxMarks": 0, "showCorrectAnswer": true, "marks": 0, "warningType": "none"}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "showFeedbackIcon": true}], "variables": {"v": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..3)", "name": "v", "description": ""}, "f1": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"<p>If&nbsp;a function $f: \\\\mathbb{R} \\\\to \\\\mathbb{R}$ is continuous at $c \\\\in \\\\mathbb{R}$, then it is differentiable at $c$.</p>\"", "name": "f1", "description": ""}, "ch2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(u=1,tr4,if(u=2,tr5,tr6))", "name": "ch2", "description": ""}, "u": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..3)", "name": "u", "description": ""}, "f2": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"<p><span>If a function $f: \\\\mathbb{R} \\\\to \\\\mathbb{R}$ is continuous </span><span>on $(a,b)$ and $f(a) &lt; \\\\gamma &lt; f(b)$, then $f(c)=\\\\gamma$ for some $c \\\\in (a,b)$</span><span>.</span></p>\"", "name": "f2", "description": ""}, "h": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "name": "h", "description": ""}, "tr1": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"<p><span>If&nbsp;</span><span>a function $f: \\\\mathbb{R} \\\\to \\\\mathbb{R}$ is differentiable at $c \\\\in \\\\mathbb{R}$, then it is continuous at $c$.</span></p>\"", "name": "tr1", "description": ""}, "f4": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"<p><span>If a function $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$</span><span>, then $f\\'(c)=0$</span><span>&nbsp;for some $c \\\\in (a,b)$.</span></p>\"", "name": "f4", "description": ""}, "f3": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"<p><span>Given any function defined on $[a,b]$ with&nbsp;$f(a) &lt; \\\\gamma &lt; f(b)$, then $f(c)=\\\\gamma$ for some $c \\\\in (a,b)$.</span></p>\"", "name": "f3", "description": ""}, "tr5": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"<p><span>If a function $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, &nbsp;then $f\\'(c)=\\\\dfrac{f(b)-f(a)}{b-a}$ for some $c \\\\in (a,b)$.</span></p>\"", "name": "tr5", "description": ""}, "ch1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(t=1,tr1,if(t=2,tr2,tr3))", "name": "ch1", "description": ""}, "f5": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"<p><span>If a function $f$ is differentiable on $(a,b)$, &nbsp;then $f\\'(c)=\\\\dfrac{f(b)-f(a)}{b-a}$&nbsp;</span>for some $c \\\\in (a,b)$.</p>\"", "name": "f5", "description": ""}, "tr2": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"<p>If a function $f$ is continuous on $[a,b]$ and $f(a) &lt; \\\\gamma &lt; f(b)$, then $f(c)=\\\\gamma$ for some $c \\\\in (a,b)$.</p>\"", "name": "tr2", "description": ""}, "f": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..3)", "name": "f", "description": ""}, "tr3": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"<p><span>If a function $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, and if $f(a) &lt; \\\\gamma &lt; f(b)$, then $f(c)=\\\\gamma$ for some $c \\\\in (a,b)$.</span></p>\"", "name": "tr3", "description": ""}, "f6": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"<p><span>If a function $f$ is continuous and differentiable on $(a,b)$, &nbsp;then $f\\'(c)=\\\\dfrac{f(b)-f(a)}{b-a}$&nbsp;</span><span>for some $c \\\\in (a,b)$.</span></p>\"", "name": "f6", "description": ""}, "tr6": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"<p><span>If a function $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, and if $f\\'(x) &gt;0$ for all $x \\\\in (a,b)$, then $f(b)&gt;f(a)$.</span></p>\"", "name": "tr6", "description": ""}, "ch4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(f=1,f4,if(f=2,f5,f6))", "name": "ch4", "description": ""}, "tr4": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"<p><span>If a function $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, and if $f(a) =&nbsp;f(b)$, then $f\\'(c)=0$ for some $c \\\\in (a,b)$.</span></p>\"", "name": "tr4", "description": ""}, "ch3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(v=1,f1,if(v=2,f2,f3))", "name": "ch3", "description": ""}, "g": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..3)", "name": "g", "description": ""}, "t": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..3)", "name": "t", "description": ""}}, "ungrouped_variables": ["f1", "f2", "f3", "f4", "f5", "f6", "t", "tr1", "u", "tr2", "tr3", "tr4", "tr5", "tr6", "g", "f", "h", "ch1", "ch2", "ch3", "ch4", "v"], "variable_groups": [], "functions": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>Answer the following question on continuity and differentiability. Note that every correct answer is worth 1 mark, but every wrong answer loses a mark.</p>", "tags": ["checked2015", "continuous", "convergence", "convergent sequences", "limits", "sequence", "sequences"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Multiple response question (2 correct out of 4) covering properties of continuity and differentiability. Selection of questions from a pool.</p>\n<p>Can choose true and false for each option. Also in one test run the second choice was incorrectly entered, rest correct,&nbsp; but the feedback indicates that the third was wrong.</p>"}, "advice": "<p>You should be able to work out the correct answers from your notes.</p>"}, {"name": "True/false statements about continuity at a point", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"del": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random('$\\\\epsilon$','$\\\\chi$','$\\\\rho$','$\\\\omega$')", "name": "del", "description": ""}, "ep": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random('$\\\\alpha$','$\\\\beta$','$\\\\gamma$','$\\\\delta$')", "name": "ep", "description": ""}}, "ungrouped_variables": ["del", "ep"], "functions": {}, "parts": [{"showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "<p>Choose the correct definitions of continuity at $x_0$ from the following:</p>\n<p></p>\n<p>[[0]]</p>", "unitTests": [], "sortAnswers": false, "scripts": {}, "gaps": [{"displayType": "checkbox", "showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "matrix": [2, 0, 0, 0, 2, 0, 2], "choices": ["<p>For every $\\var{ep} \\gt 0$ , there exists a $\\var{del} \\gt 0$ such that $\\left |f(x)-f(x_0)\\right | \\lt \\var{ep}$ whenever $|x-x_0| \\lt \\var{del}$ and $x \\in I$.</p>", "<p>For every $\\var{del} \\gt 0$ , there exists a $\\var{ep} \\gt 0$ such that $\\left |f(x)-f(x_0)\\right | \\lt \\var{ep}$ whenever $|x-x_0| \\lt \\var{del}$ and $x \\in I$.</p>", "<p>For every $\\var{ep} \\gt 0$ , there exists a $\\var{del} \\gt 0$ such that $\\left |x-x_0\\right | \\lt \\var{ep}$ whenever $|f(x)-f(x_0)| \\lt \\var{del}$ and $x \\in I$.</p>", "<p>There exists $\\var{ep} \\gt 0$ , such that for every $\\var{del} \\gt 0$, $\\left |f(x)-f(x_0)\\right | \\lt \\var{ep}$ whenever $|x-x_0| \\lt \\var{del}$ and $x \\in I$.</p>", "<p>For every $\\var{del} \\gt 0$ , there exists a $\\var{ep} \\gt 0$ such that $\\left |f(x)-f(x_0)\\right | \\lt \\var{del}$ whenever $|x-x_0| \\lt \\var{ep}$ and $x \\in I$.</p>", "<p>There exists $\\var{del} \\gt 0$ , such that for every $\\var{ep} \\gt 0$, $\\left |f(x)-f(x_0)\\right | \\lt \\var{ep}$ whenever $|x-x_0| \\lt \\var{del}$ and $x \\in I$.</p>", "<p>$\\lim_{x \\to x_0}f(x)=f(x_0)$.</p>"], "unitTests": [], "distractors": ["", "", "", "", "", "", ""], "variableReplacements": [], "type": "m_n_2", "maxAnswers": 0, "shuffleChoices": true, "showFeedbackIcon": true, "scripts": {}, "maxMarks": 0, "minAnswers": 0, "minMarks": 0, "showCellAnswerState": true, "variableReplacementStrategy": "originalfirst", "displayColumns": 1, "marks": 0}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "showFeedbackIcon": true}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>Let $\\mathbb{R}$ be the set of real numbers.</p>\n<p>Let $x_0$ be a point in the open interval $I \\subset \\mathbb{R}$ and let $f:I &nbsp;\\rightarrow \\mathbb{R}$ be a function.</p>\n<p>What does it mean to say that $f$ is continuous at $x_0$?&nbsp;</p>\n<p></p>", "tags": ["checked2015", "continuity", "continuity at a point", "continuous functions", "definition of continuity"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": ""}, "advice": ""}, {"name": "When does a sequence get within $d$ of its limit?", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s1"}, "u2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(a/c > b/d,t2,t1)", "description": "<p>Incorrect answer</p>", "name": "u2"}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "chcop(c,c)", "description": "", "name": "d"}, "w1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(u1=t1,v1,v2)", "description": "", "name": "w1"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(b1*c=a*d,b1+1,b1)", "description": "", "name": "b"}, "t2": {"templateType": "string", "group": "Ungrouped variables", "definition": "\"Decreasing\"", "description": "", "name": "t2"}, "t3": {"templateType": "string", "group": "Ungrouped variables", "definition": "\"Neither\"", "description": "", "name": "t3"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..20)", "description": "", "name": "a"}, "mono": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(a/c>b/d,1,2)", "description": "<p>If a/c &gt; b/d, the sequence is increasing. If a/c &lt; b/d, the sequence is decreasing. a,b,c,d are chosen so that $\\dfrac{a}{c} \\neq \\dfrac{b}{d}$.</p>", "name": "mono"}, "b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(2..9)", "description": "", "name": "b1"}, "t1": {"templateType": "string", "group": "Ungrouped variables", "definition": "\"Increasing\"", "description": "", "name": "t1"}, "r": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2,3,4,5,6)", "description": "", "name": "r"}, "n": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(isint(tval), tval +1,ceil(tval))", "description": "", "name": "n"}, "v2": {"templateType": "string", "group": "Ungrouped variables", "definition": "\"decreasing\"", "description": "", "name": "v2"}, "v1": {"templateType": "string", "group": "Ungrouped variables", "definition": "\"increasing\"", "description": "", "name": "v1"}, "u1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(a/c > b/d,t1,t2)", "description": "<p>Correct answer</p>", "name": "u1"}, "tval": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sqrt((1 / c) * ((10 ^ r * abs(b * c -(a * d))) / c -d))", "description": "", "name": "tval"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "chcop(a,a)", "description": "", "name": "c"}}, "ungrouped_variables": ["a", "c", "b", "s1", "b1", "d", "r", "n", "tval", "mono", "t1", "t2", "t3", "u1", "u2", "v1", "v2", "w1"], "functions": {"chcop": {"type": "number", "language": "jme", "definition": "if(gcd(a,b)=1,b,chcop(a,random(1..20)))", "parameters": [["a", "number"], ["b", "number"]]}}, "parts": [{"customMarkingAlgorithm": "", "showCorrectAnswer": true, "prompt": "<p>Find the limit $\\ell$ of $\\{x_n\\}$. Input as a fraction or an integer.</p>\n<p>Limit $\\ell=$ [[0]]</p>", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"answer": "{a}/{c}", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "vsetRangePoints": 5, "expectedVariableNames": [], "showPreview": true, "checkVariableNames": false, "checkingType": "absdiff", "vsetRange": [0, 1], "showFeedbackIcon": true, "scripts": {}, "marks": 2, "type": "jme", "unitTests": [], "checkingAccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "failureRate": 1}], "type": "gapfill", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "sortAnswers": false}, {"customMarkingAlgorithm": "", "showCorrectAnswer": true, "prompt": "<p>Find the least integer $N$ such that</p>\n<p>\\[\\left|{x_n -\\ell}\\right| &lt;&nbsp;10 ^ { -\\var{r}}, \\quad \\text{for } n \\geq N\\]</p>\n<p>Least $N=$ [[0]]</p>", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "showCorrectAnswer": true, "minValue": "{n}", "maxValue": "{n}", "unitTests": [], "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "scripts": {}, "notationStyles": ["plain", "en", "si-en"], "type": "numberentry", "variableReplacementStrategy": "originalfirst", "extendBaseMarkingAlgorithm": true, "correctAnswerFraction": false, "variableReplacements": [], "marks": "8", "showFeedbackIcon": true}], "type": "gapfill", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "sortAnswers": false}, {"displayType": "radiogroup", "choices": ["<p>{u1}</p>", "<p>{u2}</p>", "<p>{t3}</p>"], "showCorrectAnswer": true, "displayColumns": "3", "unitTests": [], "prompt": "<p>Which one of the following describes $\\{x_n\\}$?</p>", "customMarkingAlgorithm": "", "distractors": ["", "", ""], "variableReplacements": [], "shuffleChoices": true, "showFeedbackIcon": true, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "matrix": ["2", 0, 0], "marks": 0}], "statement": "<p>Let</p>\n<p>\\[x_n=\\simplify[std]{({a}n^2+{b})/({c}n^2+{d})}, \\quad n=1,2,3, \\ldots\\]</p>", "tags": ["checked2015", "convergence of a sequence", "limit", "limit of a sequence", "Limits", "limits", "query", "sequences", "taking the limit", "tested1", "udf"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>$x_n=\\frac{an^2+b}{cn^2+d}$.&nbsp;Find the least integer $N$ such that $\\left|x_n -\\frac{a}{c}\\right| &lt; 10 ^{-r},\\;n\\geq N$, $2\\leq r \\leq 6$. Determine whether the sequence is increasing, decreasing or neither.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<h4>a)&nbsp;</h4>\n<p>To begin with, the limit $\\ell$ is obtained by dividing top and bottom by $n^2$:</p>\n<p>\\[\\simplify[std]{({a}n^2+{b})/({c}n^2+{d})-{a}/{c}}&nbsp;= \\simplify[std]{({a}+{b}/n^2) /({c}+{d}/n^2)} \\to \\simplify[std]{({a})/({c})}\\] as $n \\to \\infty$, so $\\displaystyle \\ell= \\simplify[std]{{a}/{c}}$.</p>\n<h4 style=\"font-size: medium;\">b)</h4>\n<h4 style=\"font-size: medium;\">To find the least $N$ such that all terms from the $N$th are less than $10^{\\var{-r}}$ from the limit we proceed as follows:</h4>\n<p>\\begin{align}<br/>\\left|\\simplify[std]{x_n -({a} / {c})}\\right| &lt; 10 ^ { -\\var{r}} &amp;\\iff \\left|\\simplify[std]{({a}n^2+{b})/({c}n^2+{d})-{a}/{c}}\\right| &lt; 10 ^ { -\\var{r}} \\\\ <br/>&amp;\\iff \\simplify[std]{abs({b*c-a*d})/({c^2}n^2+{c*d})} &lt;10 ^ { -\\var{r}}<br/>\\end{align}</p>\n<p>(We can get rid of the absolute value in the denominator as $\\simplify[std]{{c^2}n^2+{c*d}} \\gt 0$, $\\forall n=1,2,3,\\ldots$)</p>\n<p>Rearranging this last inequality by multiplying both sides by $(\\simplify[std]{{c^2}n^2+{c*d}})10^{\\var{r}}$ (this is positive and so the inequality does not reverse), we get:</p>\n<p>\\[\\simplify[std]{{c^2}n^2+{c*d}} &gt; \\var{10^r*abs(b*c-a*d)} \\iff n^2 &gt; \\frac{1}{\\var{c^2}}\\left(\\simplify[std]{{10^r*abs(b*c-a*d)}-{c*d}}\\right)=\\var{tval^2} \\iff n&gt; \\var{tval}\\]</p>\n<p>Hence the least integer value is given by $N=\\var{N}$.</p>\n<h4>c)</h4>\n<p>Given $x_n = \\dfrac{an^2+b}{cn^2+d}, c \\gt 0, d\\gt 0$ it can be shown that $x_n \\leq x_{n+1} \\iff \\dfrac{b}{d} \\leq \\dfrac{a}{c}$. Here $\\dfrac{b}{d}=\\dfrac{\\var{b}}{\\var{d}}$ and $\\dfrac{a}{c}=\\dfrac{\\var{a}}{\\var{c}}$. Therefore the sequence will be increasing if $\\dfrac{\\var{b}}{\\var{d}} \\leq \\dfrac{\\var{a}}{\\var{c}} $ and decreasing if $\\dfrac{\\var{b}}{\\var{d}} \\geq \\dfrac{\\var{a}}{\\var{c}} $. Hence the sequence is $\\var{w1}$.</p>"}, {"name": "6. Step Function", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Nicholas Barker", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1915/"}], "rulesets": {}, "preamble": {"css": "", "js": ""}, "variablesTest": {"maxRuns": 100, "condition": ""}, "statement": "<p>The function $g(x,y)$ is a step function taking the following values on each of the rectangles shown below.</p>\n<p>{domain()}</p>\n<p>Calculate the integral</p>\n<p>\\[ \\iint_D g(x,y) dx dy \\]</p>\n<p>of $g$&nbsp;over the whole rectangle $D=[0,{\\var{MX()}}]\\times[0,{\\var{N[y]}}]$.</p>", "advice": "<p>To calculate the integral of the first rectangle $[0, \\var{M[1]}]\\times[0, \\var{N[1]}]$ we just need to find its area and multiply by the value of $g$ inside it, because $g$ is constant.</p>\n<p>\\[ I_{1,1} = \\int_0^{ \\var{N[1]}}\\int_0^{ \\var{M[1]}} g(x,y)\\, \\mathrm{d}x\\,\\mathrm{d}y =&nbsp; \\var{M[1]} \\cdot \\var{N[1]} \\cdot \\var{Q[1][1]} = \\var{M[1]*N[1]*Q[1][1]} \\]</p>\n<p>To integrate over the whole grid we integrate over every other rectangle and add all the results together.</p>\n<p>\\begin{align}<br/>I =&amp; I_{1,1} &amp;&amp;+ I_{1,2} &amp;&amp;+... \\\\<br/>&amp; \\int_0^{\\var{N[1]}}\\int_0^{\\var{M[1]}} g(x,y)\\, \\mathrm{d}x\\,\\mathrm{d}y &amp;&amp;+ \\int_0^{\\var{N[1]}}\\int_{\\var{M[1]}}^{\\var{M[2]}} g(x,y)\\, \\mathrm{d}x\\,\\mathrm{d}y &amp;&amp;+ ... \\\\<br/>=&amp; \\var{M[1]} \\cdot \\var{N[1]} \\cdot&nbsp;\\var{Q[1][1]}&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&amp;&amp;+ ( \\var{M[2]}-\\var{M[1]}) \\cdot \\var{N[1]} \\cdot&nbsp;\\var{Q[1][2]}&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&amp;&amp;+ ... \\\\<br/>=&amp; \\var{M[1]*N[1]*Q[1][1]}&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&amp;&amp;+ \\simplify{{(M[2]-M[1])*N[1]*Q[1][2]}}&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&amp;&amp;+ ... \\\\<br/> =&amp; {\\var{ans()}}<br/>\\end{align}</p>", "parts": [{"variableReplacements": [], "correctAnswerFraction": false, "maxValue": "{ans()}", "mustBeReduced": false, "showFeedbackIcon": true, "scripts": {}, "correctAnswerStyle": "plain", "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "marks": 1, "variableReplacementStrategy": "originalfirst", "minValue": "{ans()}", "allowFractions": false}], "variable_groups": [{"name": "Spacing of columns/rows", "variables": ["sm", "sn", "m1", "m2", "m3", "m4", "n1", "n2", "n3", "n4"]}], "ungrouped_variables": ["x", "y", "M", "N", "p", "Q"], "variables": {"m1": {"definition": "random(sm)", "name": "m1", "description": "", "group": "Spacing of columns/rows", "templateType": "anything"}, "n4": {"definition": "N3+random(sn)", "name": "n4", "description": "", "group": "Spacing of columns/rows", "templateType": "anything"}, "n2": {"definition": "N1+random(sn)", "name": "n2", "description": "", "group": "Spacing of columns/rows", "templateType": "anything"}, "m4": {"definition": "M3+random(sm)", "name": "m4", "description": "", "group": "Spacing of columns/rows", "templateType": "anything"}, "p": {"definition": "0..9#1", "name": "p", "description": "", "group": "Ungrouped variables", "templateType": "range"}, "n3": {"definition": "N2+random(sn)", "name": "n3", "description": "", "group": "Spacing of columns/rows", "templateType": "anything"}, "Q": {"definition": "matrix([0,random(p),random(p),random(p),random(p)],\n       [0,random(p),random(p),random(p),random(p)],\n       [0,random(p),random(p),random(p),random(p)],\n       [0,random(p),random(p),random(p),random(p)],\n       [0,0,0,0,0])", "name": "Q", "description": "<p>&nbsp;x</p>", "group": "Ungrouped variables", "templateType": "anything"}, "m3": {"definition": "M2+random(sm)", "name": "m3", "description": "", "group": "Spacing of columns/rows", "templateType": "anything"}, "m2": {"definition": "M1+random(sm)", "name": "m2", "description": "", "group": "Spacing of columns/rows", "templateType": "anything"}, "M": {"definition": "vector(0,m1,m2,m3,m4)", "name": "M", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "N": {"definition": "vector(0,n1,n2,n3,n4)", "name": "N", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "y": {"definition": "random((5-X)..(6-X) except 1)", "name": "y", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "x": {"definition": "random(2..4#1)", "name": "x", "description": "", "group": "Ungrouped variables", "templateType": "randrange"}, "n1": {"definition": "random(sn)", "name": "n1", "description": "", "group": "Spacing of columns/rows", "templateType": "anything"}, "sn": {"definition": "1..5#1", "name": "sn", "description": "", "group": "Spacing of columns/rows", "templateType": "range"}, "sm": {"definition": "2..5#1", "name": "sm", "description": "", "group": "Spacing of columns/rows", "templateType": "range"}}, "metadata": {"licence": "None specified", "description": "<p>I created this question, and every other question in Multiple Integration, for my dissertation `Computer-Aided Assessment of Multiple Integration'.</p>"}, "tags": [], "functions": {"ans": {"language": "javascript", "definition": "var X = Numbas.jme.unwrapValue(scope.variables.x);\nvar Y = Numbas.jme.unwrapValue(scope.variables.y);\nvar Q = Numbas.jme.unwrapValue(scope.variables.q);\nvar M = Numbas.jme.unwrapValue(scope.variables.m);\nvar N = Numbas.jme.unwrapValue(scope.variables.n);\n\nans = 0;\nfor (i = 1; i < X+1; i++) {\n  for (j = 1; j < Y+1; j++) {\n    ans = (M[i]-M[i-1])*(N[j]-N[j-1])*Q[j][i] + ans\n  }\n}\n\nreturn ans;", "parameters": [], "type": "number"}, "NY": {"language": "javascript", "definition": "var Y = Numbas.jme.unwrapValue(scope.variables.y);\nvar N = Numbas.jme.unwrapValue(scope.variables.n);\n\nNY = N[Y];\n\nreturn NY;", "parameters": [], "type": "number"}, "MX": {"language": "javascript", "definition": "var X = Numbas.jme.unwrapValue(scope.variables.x);\nvar M = Numbas.jme.unwrapValue(scope.variables.m);\n\nMX = M[X];\n\nreturn MX;", "parameters": [], "type": "number"}, "domain": {"language": "javascript", "definition": "var X = Numbas.jme.unwrapValue(scope.variables.x);\nvar Y = Numbas.jme.unwrapValue(scope.variables.y);\nvar M = Numbas.jme.unwrapValue(scope.variables.m);\nvar N = Numbas.jme.unwrapValue(scope.variables.n);\n\nM[X+1]=0; M[X+2]=0;\nN[Y+1]=0; N[Y+2]=0;\n\nvar makeboard = Numbas.extensions.jsxgraph.makeBoard(\n  '250px','250px', \n  {boundingBox:[-3/2,N[Y]+1/2,M[X]+3/2,-3/4], \n   axis:true, \n   grid:true\n  } ); // define how the graph is displayed\n\nvar board = makeboard.board; // generate the graph\n\nvar column1 = board.create('curve',\n                 [function(y){return M[1];},\n                  function(y){return y;},\n                  0, N[Y]]);\nvar column2 = board.create('curve',\n                 [function(y){return M[2];},\n                  function(y){return y;},\n                  0, N[Y]]);\nvar column3 = board.create('curve',\n                 [function(y){return M[3];},\n                  function(y){return y;},\n                  0, N[Y]]);\nvar column4 = board.create('curve',\n                 [function(y){return M[X];},\n                  function(y){return y;},\n                  0, N[Y]]);\n\nvar row1 = board.create('curve',\n                 [function(y){return y;},\n                  function(y){return N[1];},\n                  0, M[X]]);\nvar row2 = board.create('curve',\n                 [function(y){return y;},\n                  function(y){return N[2];},\n                  0, M[X]]);\nvar row3 = board.create('curve',\n                 [function(y){return y;},\n                  function(y){return N[3];},\n                  0, M[X]]);\nvar row4 = board.create('curve',\n                 [function(y){return y;},\n                  function(y){return N[Y];},\n                  0, M[X]]);\n\nvar Q = Numbas.jme.unwrapValue(scope.variables.q);\nfor (i = 1; i < X+1; i++) {\n  for (j = 1; j < Y+1; j++) {\n    p = Q[j][i] // random integer in [-9,9]\n    var points = board.create('point', [-2,0],\n          {label:{fontSize:15,\n                  offset:[(125/(M[X]+3))*(4+ M[i-1]+M[i] + 0.10*(M[i-1]-M[i])),\n                          (125/(N[Y]+5/4))*(N[j-1]+N[j])]}, name:p, fixed:true});\n  }\n}\n\nreturn makeboard; // update the board", "parameters": [], "type": "html"}}, "type": "question"}, {"name": "Sequences: Limit Laws", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}], "functions": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>The following questions deal with the limit laws of sequences.</p>\n<p></p>", "preamble": {"js": "", "css": ""}, "ungrouped_variables": ["A", "B", "C", "D", "L", "f", "g", "list", "n0", "h", "j"], "parts": [{"showCorrectAnswer": true, "scripts": {}, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "gaps": [{"showCorrectAnswer": true, "scripts": {}, "variableReplacements": [], "showFeedbackIcon": true, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "correctAnswerStyle": "plain", "maxValue": "(A+f*B-C*D)/(L+g)", "notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "marks": 1, "correctAnswerFraction": true, "allowFractions": true, "minValue": "(A+f*B-C*D)/(L+g)", "mustBeReducedPC": 0}], "marks": 0, "prompt": "<p>Given that</p>\n<p>$\\displaystyle\\lim_{n\\rightarrow\\infty}a_n=\\var{A}$,&nbsp;<br/>$\\displaystyle\\lim_{n\\rightarrow\\infty}b_n=\\var{B}$,&nbsp;<br/>$\\displaystyle\\lim_{n\\rightarrow\\infty}c_n=\\var{C}$, <br/>$\\displaystyle\\lim_{n\\rightarrow\\infty}d_n=\\var{D}$ and <br/>$\\displaystyle\\lim_{n\\rightarrow\\infty}l_n=\\var{L}$.</p>\n<p>Determine the following limit:</p>\n<p style=\"text-align: center;\">$\\displaystyle\\lim_{n\\rightarrow\\infty} \\left(\\frac{a_n+\\var{f}b_n-c_nd_n}{l_n+\\var{g}}\\right)=$&nbsp;[[0]]</p>", "variableReplacements": [], "type": "gapfill"}, {"showCorrectAnswer": true, "scripts": {}, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "gaps": [{"showCorrectAnswer": true, "distractors": ["", "", "", ""], "scripts": {}, "variableReplacements": [], "showFeedbackIcon": true, "matrix": [0, 0, 0, "1"], "minMarks": 0, "choices": ["<p>It diverges.</p>", "<p>It converges to some unknown number.</p>", "<p>There is insufficient&nbsp;information to tell if it converges or diverges.</p>", "<p>It converges to {h}.</p>"], "displayType": "dropdownlist", "variableReplacementStrategy": "originalfirst", "type": "1_n_2", "marks": 0, "maxMarks": 0, "displayColumns": 0, "shuffleChoices": true}], "marks": 0, "prompt": "<p>Suppose $x_n\\le y_n \\le z_n$ for all $n\\ge \\var{n0}$, and $\\displaystyle\\lim_{n\\rightarrow\\infty}x_n=\\lim_{n\\rightarrow\\infty}z_n=\\var{h}$. Then what can be said about the sequence $\\{y_n\\}$?</p>\n<p>[[0]]</p>", "variableReplacements": [], "type": "gapfill"}, {"showCorrectAnswer": true, "scripts": {}, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "gaps": [{"showCorrectAnswer": true, "distractors": ["", "", "", "", ""], "scripts": {}, "variableReplacements": [], "showFeedbackIcon": true, "matrix": [0, 0, 0, "1", 0], "minMarks": 0, "choices": ["<p>It diverges.</p>", "<p>It converges to some unknown number.</p>", "<p>There is insufficient&nbsp;information to tell if it converges or diverges.</p>", "<p>It converges to {j}.</p>", "<p>It converges to {j+0.5}.</p>"], "displayType": "dropdownlist", "variableReplacementStrategy": "originalfirst", "type": "1_n_2", "marks": 0, "maxMarks": 0, "displayColumns": 0, "shuffleChoices": true}], "marks": 0, "prompt": "<p>Suppose $\\lim_{x\\rightarrow\\infty}f(x)=\\var{j}$ and $f(n)=a_n$ when $n$ is an integer. Then what can be said about the sequence $\\{a_n\\}$?</p>\n<p>[[0]]</p>", "variableReplacements": [], "type": "gapfill"}], "rulesets": {}, "variables": {"C": {"templateType": "anything", "name": "C", "group": "Ungrouped variables", "description": "", "definition": "list[2]"}, "D": {"templateType": "anything", "name": "D", "group": "Ungrouped variables", "description": "", "definition": "list[3]"}, "h": {"templateType": "anything", "name": "h", "group": "Ungrouped variables", "description": "", "definition": "list[7]"}, "j": {"templateType": "anything", "name": "j", "group": "Ungrouped variables", "description": "", "definition": "list[8]+0.5"}, "A": {"templateType": "anything", "name": "A", "group": "Ungrouped variables", "description": "", "definition": "list[0]"}, "L": {"templateType": "anything", "name": "L", "group": "Ungrouped variables", "description": "", "definition": "list[4]"}, "B": {"templateType": "anything", "name": "B", "group": "Ungrouped variables", "description": "", "definition": "list[1]"}, "g": {"templateType": "anything", "name": "g", "group": "Ungrouped variables", "description": "", "definition": "abs(list[6])"}, "f": {"templateType": "anything", "name": "f", "group": "Ungrouped variables", "description": "", "definition": "abs(list[5])"}, "list": {"templateType": "anything", "name": "list", "group": "Ungrouped variables", "description": "", "definition": "shuffle(-12..12 except 0)"}, "n0": {"templateType": "anything", "name": "n0", "group": "Ungrouped variables", "description": "", "definition": "random(10..1000#10)"}}, "variable_groups": [], "advice": "<p>a)&nbsp;</p>\n<p>The following steps through the application of the limit laws to our question</p>\n<p>$\\displaystyle\\begin{align}\\lim_{n\\rightarrow\\infty} \\left(\\frac{a_n+\\var{f}b_n-c_nd_n}{l_n+\\var{g}}\\right)&amp;=\\frac{\\lim_{n\\rightarrow\\infty}\\left(a_n+\\var{f}b_n-c_nd_n\\right)}{\\lim_{n\\rightarrow\\infty}\\left(l_n+\\var{g}\\right)}\\\\&amp;=\\frac{\\lim_{n\\rightarrow\\infty}(a_n)+\\lim_{n\\rightarrow\\infty}(\\var{f}b_n)-\\lim_{n\\rightarrow\\infty}(c_nd_n)}{\\lim_{n\\rightarrow\\infty}(l_n)+\\lim_{n\\rightarrow\\infty}(\\var{g})}\\\\&amp;=\\frac{\\lim_{n\\rightarrow\\infty}(a_n)+\\var{f}\\lim_{n\\rightarrow\\infty}(b_n)-\\lim_{n\\rightarrow\\infty}(c_n)\\lim_{n\\rightarrow\\infty}(d_n)}{\\lim_{n\\rightarrow\\infty}(l_n)+\\lim_{n\\rightarrow\\infty}(\\var{g})}\\\\&amp;=\\frac{\\var{A}+\\var{f}(\\var{B})-(\\var{C})(\\var{D})}{\\var{L}+\\var{g}}\\\\&amp;=\\simplify[fractionNumbers,simplifyFractions,unitDenominator]{{A+f*B-C*D}/{L+g}}\\end{align}$</p>\n<p></p>\n<p>b)&nbsp;</p>\n<p>The middle sequence,&nbsp;$\\{y_n\\}$, is squeezed by the sequence below, $\\{x_n\\}$, and the sequence above, $\\{z_n\\}$, which both converge to the same limit, therefore $\\{y_n\\}$ must also converge to the same limit. This is known as the squeeze theorem (for sequences).</p>\n<p></p>\n<p>c)&nbsp;</p>\n<p>The only difference between a function converging and a sequence converging is that the sequence is only defined for integers. So if&nbsp;the&nbsp;value of&nbsp;$a_n$ is the same as $f(n)$ and $f(x)$ converges to $\\var{j}$, then it stands to reason that the sequence $\\{a_n\\}$ would also converge to $\\var{j}$.</p>", "tags": [], "metadata": {"licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International", "description": "<p>Basic question on limit laws and the squeeze theorem for sequences.</p>"}, "type": "question"}, {"name": "Series: divergence test", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}], "ungrouped_variables": ["text", "seed", "limit", "existence", "cseed", "a", "d", "coeff", "b", "c", "f", "g", "h", "alt", "cexpression", "con_to_zero", "div_to_inf", "jump_around", "con_to_non_zero", "start"], "functions": {}, "rulesets": {}, "tags": [], "advice": "<p>The divergence test says that&nbsp;<strong>if</strong> the sequence of $t_k$ diverges or converges to a non-zero number<em>&nbsp;</em><strong>then</strong>&nbsp;the series $\\sum_{k=a}^\\infty t_k$ diverges. Notice it does not tell us anything about the series if the sequence of $t_k$ converges to $0$. Another way to think about this is, for a series to have any <em><g class=\"gr_ gr_62 gr-alert gr_gramm gr_inline_cards gr_run_anim Style multiReplace\" id=\"62\" data-gr-id=\"62\">chance&nbsp;</g></em><g class=\"gr_ gr_62 gr-alert gr_gramm gr_inline_cards gr_disable_anim_appear Style multiReplace\" id=\"62\" data-gr-id=\"62\"> of</g> converging then we require&nbsp;its terms to converge to zero.&nbsp;You might see the divergence test written more concisely as \\[\\lim_{k\\rightarrow\\infty} t_k\\ne 0 \\implies \\sum_{k=0}^\\infty t_k \\,\\,\\text{ diverges.}\\]</p>\n<p>a)&nbsp;</p>\n<p><span data-jme-visible=\"seed=1\">Since each term in the series is getting {text} in absolute value the sequence can never 'settle down' or 'approach' a finite number. In other words, the sequence diverges (does not converge) and so the divergence test implies that the series diverges.</span></p>\n<p><span data-jme-visible=\"seed=-1\">Since each term in the series is getting {text} in absolute value the sequence 'settles down'&nbsp;and 'approaches' zero. In other words, the sequence&nbsp;converges to zero and so the divergence test doesn't actually tell us anything about the series (it may or may not converge).</span></p>\n<p></p>\n<p>b)</p>\n<p><span data-jme-visible=\"existence=0\">Since the limit of the sequence does not exist, the sequence diverges (diverges just means does not converge) and so the divergence test tells us that the series diverges.</span></p>\n<p><span data-jme-visible=\"existence=1 and seed=-1\">Since the limit of the sequence&nbsp;is a non-zero number ($\\var[fractionNumbers]{limit}$) the divergence test&nbsp;tells us that the series diverges.</span></p>\n<p><span data-jme-visible=\"existence=1 and seed=1\">Since the limit of the sequence&nbsp;is zero, the divergence test doesn't actually tell us anything about whether the series converges or diverges.</span></p>\n<p></p>\n<p>c)</p>\n<p>To use the divergence test we need to determine whether the sequence of terms approach zero or not.&nbsp;Consider &nbsp;&nbsp;</p>\n<p style=\"text-align: center;\">{cexpression} as $k\\rightarrow \\infty$.</p>\n<p></p>\n<p></p>\n<p><span data-jme-visible=\"cseed='inf' and alt=0\">As $k$ gets larger and larger our terms get larger and larger.&nbsp;This means that&nbsp;the limit does not exist (or sometimes people prefer to say the limit equals infinity) and so the divergence test tells us that the series diverges.</span></p>\n<p></p>\n<p><span data-jme-visible=\"cseed='inf' and alt=1\">As $k$ gets larger and larger our terms get larger and larger in absolute value but alternate in sign.&nbsp;This means that&nbsp;the limit does not exist and so the divergence test tells us that the series diverges.</span></p>\n<p></p>\n<p><span data-jme-visible=\"cseed='jump'\">As $k$ gets larger and larger our terms jump around, they get positive, negative, smaller, larger... The terms do not approach a constant.&nbsp;This means that&nbsp;the limit does not exist and so the divergence test tells us that the series diverges.</span></p>\n<p></p>\n<p><span data-jme-visible=\"cseed='zero'\">As $k$ gets larger and larger our terms approach zero. The larger $k$ gets, the closer to zero the terms become. This means that the sequence converges to $0$ and so the divergence test does not tell us anything about whether the series converges or diverges.</span></p>\n<p></p>\n<p><span data-jme-visible=\"cseed='non_zero'\">As $k$ gets larger and larger our terms approach a non-zero constant. The larger $k$ gets, the closer to&nbsp;this non-zero constant the terms become. This means that the sequence converges to this non-zero constant and so the divergence test tells us that the series diverges.</span></p>\n<p></p>", "preamble": {"css": "", "js": ""}, "variable_groups": [], "parts": [{"distractors": ["", "", ""], "scripts": {}, "prompt": "<p>A series is such that each term&nbsp;has an absolute value that is&nbsp;{text} than the last. What does the <strong>divergence test</strong> tell us about this series?</p>", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "marks": 0, "shuffleChoices": false, "minMarks": 0, "displayColumns": 0, "choices": ["<p>This series converges.</p>", "<p>This series diverges.</p>", "<p>It doesn't tell us anything.</p>"], "variableReplacements": [], "showCorrectAnswer": true, "displayType": "dropdownlist", "type": "1_n_2", "matrix": [0, "if(seed=1,1,0)", "if(seed=-1,1,0)"], "maxMarks": 0}, {"distractors": ["", "", ""], "scripts": {}, "prompt": "<p>A series,&nbsp;$\\displaystyle\\sum_{k=\\var{start}}^\\infty t_k$, is such that $\\displaystyle\\lim_{k\\rightarrow \\infty}t_k$&nbsp;<span data-jme-visible=\"existence=1\">$=\\var[fractionNumbers]{limit}$</span>&nbsp;&nbsp;<span data-jme-visible=\"existence=0\">does not exist</span>. What does the <strong>divergence test</strong> tell us about this series?</p>", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "marks": 0, "shuffleChoices": false, "minMarks": 0, "displayColumns": 0, "choices": ["<p>This series converges.</p>", "<p>This series diverges.</p>", "<p>It doesn't tell us anything.</p>"], "variableReplacements": [], "showCorrectAnswer": true, "displayType": "dropdownlist", "type": "1_n_2", "matrix": [0, "if(seed=-1,1,0)", "if(seed=1,1,0)"], "maxMarks": 0}, {"distractors": ["", "", ""], "scripts": {}, "prompt": "<p>Given the series</p>\n<p style=\"text-align: center;\">$\\displaystyle\\sum_{k=\\var{start}}^\\infty$ {cexpression}</p>\n<p>What does the <strong>divergence test</strong> tell us about this series?</p>", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "marks": 0, "shuffleChoices": false, "minMarks": 0, "displayColumns": 0, "choices": ["<p>This series converges.</p>\n<p></p>", "<p>This series diverges.</p>", "<p>It doesn't tell us anything.</p>"], "variableReplacements": [], "showCorrectAnswer": true, "displayType": "dropdownlist", "type": "1_n_2", "matrix": [0, "if(cseed='inf' or cseed='jump' or cseed='non_zero',1,0)", "if(cseed='zero',1,0)"], "maxMarks": 0}], "variables": {"cseed": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "cseed", "definition": "random('zero','non_zero','inf','jump')"}, "c": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "c", "definition": "coeff[1]"}, "g": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "g", "definition": "coeff[3]"}, "b": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "b", "definition": "coeff[0]"}, "a": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "a", "definition": "random(2..12)"}, "d": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "d", "definition": "random(-12..12 except [0,a,-a])"}, "limit": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "limit", "definition": "if(seed=1,0,random(-1/9,-1/8,-1/7,-1/5,-1/4,-1/3,-1/2,-2,-1,1/9,1/8,1/7,1/5,1/4,1/3,1/2,2,1, 1/10, -1/10, -1/100, 1/100))"}, "alt": {"group": "Ungrouped variables", "templateType": "anything", "description": "<p>1 for alternating sign, 0 for not alternating sign</p>", "name": "alt", "definition": "random(0,1)"}, "start": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "start", "definition": "random(0..5)"}, "seed": {"group": "Ungrouped variables", "templateType": "anything", "description": "<p>for part b 1= inconclusive, -1=series diverges REVERSED for part a</p>", "name": "seed", "definition": "random(-1,1)"}, "jump_around": {"group": "Ungrouped variables", "templateType": "anything", "description": "<p></p>\n<p>This requires alt to equal 1...</p>\n<p>This is really 'diverge but stay bounded'.</p>", "name": "jump_around", "definition": "[\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*{d}*arctan(k)}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*{a}}\\$\",   \n\"\\$\\\\displaystyle\\\\simplify{(-1)^k}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*({a}k+{b})/({d}k+{f})}\\$\",    \n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*({a}k^2+{b}k+{c})/({d}k^2+{f}k+{g})}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*({a}k^3+{b}k^2+{c}k+{f})/({d}k^3+{f}k^2+{g}k+{h})}\\$\", \n\"\\$\\\\displaystyle\\\\simplify{sin(k)}\\$\", \n\"\\$\\\\displaystyle\\\\simplify{cos(k)}\\$\", \n\"\\$\\\\displaystyle\\\\simplify{tan(k)}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*sin(k)}\\$\", \n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*cos(k)}\\$\", \n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*tan(k)}\\$\"\n]"}, "con_to_zero": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "con_to_zero", "definition": "if(alt=1,\n[\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k/({a}^k)}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k/(k+1)}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*{a}/k!}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*k!/(k+{a})!}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*{a}^k/k!}\\$\",   \n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*({a}k+{b})/({d}k^2+{f}k+{g})}\\$\", \n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*({a}k^2+{b}k+{c})/({d}k^3+{f}k^2+{g}k+{h})}\\$\"\n],\n  [\n\"\\$\\\\displaystyle\\\\simplify{1/({a}^k)}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{1/(k+1)}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{{a}/k!}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{k!/(k+{a})!}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{{a}^k/k!}\\$\",    \n\"\\$\\\\displaystyle\\\\simplify{({a}k+{b})/({d}k^2+{f}k+{g})}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{({a}k^2+{b}k+{c})/({d}k^3+{f}k^2+{g}k+{h})}\\$\"\n])"}, "div_to_inf": {"group": "Ungrouped variables", "templateType": "anything", "description": "<p>This is really 'diverge in absolute value to infinity'</p>", "name": "div_to_inf", "definition": "if(alt=1,\n[\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*({a}^k)}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*k}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*k!/{a}}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*(k+{a})!/k!}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*k!/{a}^k}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*({d}k^2+{f}k+{g})/({a}k+{b})}\\$\", \n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*({d}k^3+{f}k^2+{g}k+{h})/({a}k^2+{b}k+{c})}\\$\"\n],\n  [\n\"\\$\\\\displaystyle\\\\simplify{{a}^k}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{k}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{k!/{a}}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{(k+{a})!/k!}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{k!/{a}^k}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{({d}k^2+{f}k+{g})/({a}k+{b})}\\$\", \n\"\\$\\\\displaystyle\\\\simplify{({d}k^3+{f}k^2+{g}k+{h})/({a}k^2+{b}k+{c})}\\$\"\n])"}, "cexpression": {"group": "Ungrouped variables", "templateType": "anything", "description": "<p>cseed=random('zero','non_zero','inf','jump')</p>", "name": "cexpression", "definition": "if(cseed='zero',random(con_to_zero),\nif(cseed='non_zero',random(con_to_non_zero),\nif(cseed='inf',random(div_to_inf), \nif(cseed='jump',random(jump_around),''))))"}, "con_to_non_zero": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "con_to_non_zero", "definition": "if(alt=1,\n[\n\"\\$\\\\displaystyle\\\\simplify{({a}k^3+{b}k^2+{c}k+{f})/({d}k^3+{f}k^2+{g}k+{h})+(-1)^k/({a}^k)}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{({a}k^2+{b}k+{c})/({d}k^2+{f}k+{g})+(-1)^k/(k+1)}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{({a}k+{b})/({d}k+{f})+(-1)^k*{a}/k!}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{{a}+(-1)^k*k!/(k+{a})!}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{{d}*arctan(k)+(-1)^k*{a}^k/k!}\\$\",   \n\"\\$\\\\displaystyle\\\\simplify{({a}k+{b})/({d}k+{f})+(-1)^k*({a}k+{b})/({d}k^2+{f}k+{g})}\\$\", \n\"\\$\\\\displaystyle\\\\simplify{{a}+(-1)^k*({a}k^2+{b}k+{c})/({d}k^3+{f}k^2+{g}k+{h})}\\$\"\n],\n  [\n\"\\$\\\\displaystyle\\\\simplify{{d}*arctan(k)}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{{a}}\\$\",   \n\"\\$\\\\displaystyle\\\\simplify{({a}k+{b})/({d}k+{f})}\\$\",    \n\"\\$\\\\displaystyle\\\\simplify{({a}k^2+{b}k+{c})/({d}k^2+{f}k+{g})}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{({a}k^3+{b}k^2+{c}k+{f})/({d}k^3+{f}k^2+{g}k+{h})}\\$\"\n])"}, "text": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "text", "definition": "if(seed=1,'larger', 'smaller')"}, "coeff": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "coeff", "definition": "shuffle(list(-12..12) +0+0+0+0+0+0+0+0+0+0+0+0)[0..5]"}, "f": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "f", "definition": "coeff[2]"}, "existence": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "existence", "definition": "random(0,1)"}, "h": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "h", "definition": "coeff[4]"}}, "metadata": {"description": "<p>Test whether a student knows the divergence test of a series, and how to use it. Series include those that the <g class=\"gr_ gr_7 gr-alert gr_spell gr_inline_cards gr_run_anim ContextualSpelling ins-del multiReplace\" id=\"7\" data-gr-id=\"7\">diverence</g> test is inconclusive for. This question could be better in that it could go through the working of determining the limit but I&nbsp;hope to&nbsp;make a separate question which deals with that.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "variablesTest": {"maxRuns": 100, "condition": ""}, "statement": "<p>This question is about the divergence test for series.</p>", "type": "question"}, {"name": "Series: ratio test", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}], "ungrouped_variables": ["seed", "percent", "limit", "a", "d", "coeff", "b", "c", "f", "g", "h", "alt", "cseed", "cexpression", "ratio_con_to_zero", "ratio_con_to_less_than_one", "ratio_con_to_one", "ratio_con_to_big", "ratio_div_to_inf", "start"], "functions": {}, "rulesets": {}, "tags": [], "advice": "<p>The ratio test says, given a series&nbsp;$\\sum_{k=a}^\\infty t_k$</p>\n<ul>\n<li>if $\\lim_{k\\rightarrow\\infty}\\left|\\frac{t_{k+1}}{t_k}\\right|&lt;1$, then the series converges</li>\n<li>if $\\lim_{k\\rightarrow\\infty}\\left|\\frac{t_{k+1}}{t_k}\\right|&gt;1$, then the series&nbsp;diverges</li>\n<li>if $\\left|\\frac{t_{k+1}}{t_k}\\right|\\rightarrow\\infty$ as $k\\rightarrow \\infty$, then the series diverges</li>\n</ul>\n<p>Notice, if the limit doesn't exist for some other reason or is equal to $1$ the test doesn't tell us anything, some might say the test is 'inconclusive' or 'fails'.&nbsp;</p>\n<p></p>\n<p>a)&nbsp;</p>\n<p><span data-jme-visible=\"seed=-1\">Since&nbsp;$\\lim_{k\\rightarrow\\infty}\\left|\\frac{t_{k+1}}{t_k}\\right|=\\frac{\\var{percent}}{100}&lt;1$ the ratio test tells us that the series converges.</span></p>\n<p><span data-jme-visible=\"seed=1\">Since&nbsp;$\\lim_{k\\rightarrow\\infty}\\left|\\frac{t_{k+1}}{t_k}\\right|=\\frac{\\var{percent}}{100}&gt;1$ the ratio test tells us that the series diverges.</span></p>\n<p><span data-jme-visible=\"seed=0\">Since&nbsp;$\\lim_{k\\rightarrow\\infty}\\left|\\frac{t_{k+1}}{t_k}\\right|=1$ the ratio test doesn't tell us anything.</span></p>\n<p></p>\n<p>b)</p>\n<p><span data-jme-visible=\"seed=1\">Since&nbsp;$\\lim_{k\\rightarrow\\infty}\\left|\\frac{t_{k+1}}{t_k}\\right|=\\var[fractionNumbers]{limit}&lt;1$ the ratio test tells us that the series converges.</span></p>\n<p><span data-jme-visible=\"seed=0\">Since&nbsp;$\\lim_{k\\rightarrow\\infty}\\left|\\frac{t_{k+1}}{t_k}\\right|=\\var[fractionNumbers]{limit}&gt;1$ the ratio test tells us that the series diverges.</span></p>\n<p><span data-jme-visible=\"seed=-1\">Since&nbsp;$\\lim_{k\\rightarrow\\infty}\\left|\\frac{t_{k+1}}{t_k}\\right|=1$ the ratio test doesn't tell us anything.</span></p>\n<p></p>\n<p>c)</p>\n<p>To use the&nbsp;ratio test we need to determine the value that the absolute value of the ratio of consecutive terms converge to.&nbsp;</p>\n<p>Given $t_k=${cexpression} we use algebra and limit laws to determine</p>\n<p><span data-jme-visible=\"cseed=0\">\\[\\lim_{k\\rightarrow\\infty}\\frac{t_{k+1}}{t_k}=0\\]</span></p>\n<p><span data-jme-visible=\"cseed=0.5\">\\[\\lim_{k\\rightarrow\\infty}\\frac{t_{k+1}}{t_k}=\\simplify[fractionNumbers]{{1/a}}\\]</span></p>\n<p><span data-jme-visible=\"cseed=1\">\\[\\lim_{k\\rightarrow\\infty}\\frac{t_{k+1}}{t_k}=1\\]</span></p>\n<p><span data-jme-visible=\"cseed=2\">\\[\\lim_{k\\rightarrow\\infty}\\frac{t_{k+1}}{t_k}=\\var{a}\\]</span></p>\n<p></p>\n<p><span data-jme-visible=\"cseed=0 or cseed=0.5\">and therefore by the ratio test the series converges.</span></p>\n<p><span data-jme-visible=\"cseed=2\">and therefore by the ratio test the series diverges.&nbsp;</span></p>\n<p><span data-jme-visible=\"cseed=1\">and therefore the ratio test&nbsp;doesn't tell us anything.</span></p>\n<p></p>", "preamble": {"css": "", "js": ""}, "variable_groups": [], "parts": [{"distractors": ["", "", ""], "scripts": {}, "prompt": "<p>A series is such that as we get further and further along the sequence, each term&nbsp;has an absolute value&nbsp;that approaches $\\var{percent}\\%$ of the absolute value of the previous term. What does the <strong>ratio&nbsp;test</strong> tell us about this series?</p>", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "marks": 0, "shuffleChoices": false, "minMarks": 0, "displayColumns": 0, "choices": ["<p>This series converges.</p>", "<p>This series diverges.</p>", "<p>It doesn't tell us anything.</p>"], "variableReplacements": [], "showCorrectAnswer": true, "displayType": "dropdownlist", "type": "1_n_2", "matrix": ["if(seed=-1,1,0)", "if(seed=1,1,0)", "if(seed=0,1,0)"], "maxMarks": 0}, {"distractors": ["", "", ""], "scripts": {}, "prompt": "<p>A series,&nbsp;$\\displaystyle\\sum_{k=\\var{start}}^\\infty t_k$, is such that $\\displaystyle\\lim_{k\\rightarrow \\infty}\\left\\vert\\frac{t_{k+1}}{t_k}\\right\\vert $&nbsp;$=\\var[fractionNumbers]{limit}$. What does the <strong>ratio test</strong> tell us about this series?</p>", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "marks": 0, "shuffleChoices": false, "minMarks": 0, "displayColumns": 0, "choices": ["<p>This series converges.</p>", "<p>This series diverges.</p>", "<p>It doesn't tell us anything.</p>"], "variableReplacements": [], "showCorrectAnswer": true, "displayType": "dropdownlist", "type": "1_n_2", "matrix": ["if(seed=1,1,0)", "if(seed=0,1,0)", "if(seed=-1,1,0)"], "maxMarks": 0}, {"distractors": ["", "", ""], "scripts": {}, "prompt": "<p>Given the series</p>\n<p style=\"text-align: center;\">$\\displaystyle\\sum_{k=\\var{start}}^\\infty$ {cexpression}</p>\n<p>What does the <strong>ratio test</strong> tell us about this series?</p>", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "marks": 0, "shuffleChoices": false, "minMarks": 0, "displayColumns": 0, "choices": ["<p>This series converges.</p>", "<p>This series diverges.</p>", "<p>It doesn't tell us anything.</p>"], "variableReplacements": [], "showCorrectAnswer": true, "displayType": "dropdownlist", "type": "1_n_2", "matrix": ["if(cseed=0 or cseed=0.5,1,0)", "if(cseed=2,1,0)", "if(cseed=1,1,0)"], "maxMarks": 0}], "variables": {"cseed": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "cseed", "definition": "random(0,0.5,1,1,2,2)"}, "d": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "d", "definition": "random(-12..12 except [0,a,-a])"}, "b": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "b", "definition": "coeff[0]"}, "g": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "g", "definition": "coeff[3]"}, "a": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "a", "definition": "random(2..12)"}, "ratio_con_to_big": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "ratio_con_to_big", "definition": "if(alt=1,\n[\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*({a}^(k+1))}\\$\",//ratio converges to a \n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*({a}^(k+2))*k!/(k+{a})!)}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*({a}^k)*({a}k+{b})/({d}k^2+{f}k+{g})}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*({a}^(k+1))*({a}k^2+{b}k+{c})/({d}k^3+{f}k^2+{g}k+{h})}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*k*({a}^(k+2))}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*{a}^k*(k+{a})!/(k!)}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*{a}^(k+1)*({d}k^2+{f}k+{g})/({a}k+{b})}\\$\", \n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*{a}^(k+2)*({d}k^3+{f}k^2+{g}k+{h})/({a}k^2+{b}k+{c})}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*({a}^k)*arctan(k)}\\$\",  \n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*{a}^(k+1)*({a}k+{b})/({d}k+{f})}\\$\"\n]\n,\n[\n\"\\$\\\\displaystyle\\\\simplify{({a}^(k+1))}\\$\",//ratio converges to a \n\"\\$\\\\displaystyle\\\\simplify{({a}^(k+2)*k!)/(k+{a})!)}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{({a}^k)*({a}k+{b})/({d}k^2+{f}k+{g})}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{({a}^(k+1))*({a}k^2+{b}k+{c})/({d}k^3+{f}k^2+{g}k+{h})}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{k*({a}^(k+2))}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{{a}^k*(k+{a})!/(k!)}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{{a}^(k+1)*({d}k^2+{f}k+{g})/({a}k+{b})}\\$\", \n\"\\$\\\\displaystyle\\\\simplify{{a}^(k+2)*({d}k^3+{f}k^2+{g}k+{h})/({a}k^2+{b}k+{c})}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{({a}^k)*arctan(k)}\\$\",  \n\"\\$\\\\displaystyle\\\\simplify{{a}^(k+1)*({a}k+{b})/({d}k+{f})}\\$\"\n])"}, "limit": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "limit", "definition": "if(seed=1,random(1/9,1/8,1/7,1/5,1/4,1/3,1/2,1/10,1/100),if(seed=0,random(10/9,11/8,8/7,7/5,5/4,4/3,5/2,2,17/10,102/100),1))"}, "alt": {"group": "Ungrouped variables", "templateType": "anything", "description": "<p>1 for alternating sign, 0 for not alternating sign</p>", "name": "alt", "definition": "random(0,1)"}, "ratio_con_to_less_than_one": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "ratio_con_to_less_than_one", "definition": "if(alt=1,\n[\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k/({a}^k)}\\$\",//ratio converges to 1/a \n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*k!/({a}^(k+1)*(k+{a})!)}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*({a}k+{b})/({a}^(k+2)*({d}k^2+{f}k+{g}))}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*({a}k^2+{b}k+{c})/({a}^k*({d}k^3+{f}k^2+{g}k+{h}))}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*k/({a}^(k+1)}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*(k+{a})!/({a}^(k+2)*k!)}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*({d}k^2+{f}k+{g})/({a}^k*({a}k+{b}))}\\$\", \n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*({d}k^3+{f}k^2+{g}k+{h})/({a}^(k+1)*({a}k^2+{b}k+{c}))}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*arctan(k)/({a}^(k+2))}\\$\",  \n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*({a}k+{b})/({a}^k*({d}k+{f}))}\\$\"\n]\n,\n[\n\"\\$\\\\displaystyle\\\\simplify{1/({a}^k)}\\$\",//ratio converges to 1/a \n\"\\$\\\\displaystyle\\\\simplify{k!/({a}^(k+1)*(k+{a})!)}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{({a}k+{b})/({a}^(k+2)*({d}k^2+{f}k+{g}))}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{({a}k^2+{b}k+{c})/({a}^k*({d}k^3+{f}k^2+{g}k+{h}))}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{k/({a}^(k+1))}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{(k+{a})!/({a}^(k+2)*k!)}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{({d}k^2+{f}k+{g})/({a}^k*({a}k+{b}))}\\$\", \n\"\\$\\\\displaystyle\\\\simplify{({d}k^3+{f}k^2+{g}k+{h})/({a}^(k+1)*({a}k^2+{b}k+{c}))}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{arctan(k)/({a}^(k+2))}\\$\",  \n\"\\$\\\\displaystyle\\\\simplify{({a}k+{b})/({a}^k*({d}k+{f}))}\\$\"\n])"}, "coeff": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "coeff", "definition": "shuffle(list(-12..12) +0+0+0+0+0+0+0+0+0+0+0+0)[0..5]"}, "ratio_div_to_inf": {"group": "Ungrouped variables", "templateType": "anything", "description": "<p>I have omitted these but you might like to use them, it depends on what your course notes say about the ratio test...</p>", "name": "ratio_div_to_inf", "definition": "if(alt=1,\n[\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*k!/{a}}\\$\",  \n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*k!/{a}^k}\\$\"\n],\n  [\n\"\\$\\\\displaystyle\\\\simplify{k!/{a}}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{k!/{a}^k}\\$\"\n])"}, "c": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "c", "definition": "coeff[1]"}, "start": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "start", "definition": "random(0..5)"}, "seed": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "seed", "definition": "random(-1,0,1)"}, "ratio_con_to_zero": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "ratio_con_to_zero", "definition": "if(alt=1,\n[\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k/(k+1)}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*{a}/k!}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*{a}^k/k!}\\$\"\n],\n  [\n\"\\$\\\\displaystyle\\\\simplify{1/(k+1)}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{{a}/k!}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{{a}^k/k!}\\$\"\n])"}, "cexpression": {"group": "Ungrouped variables", "templateType": "anything", "description": "<p>random(0,0.5,1,2)</p>", "name": "cexpression", "definition": "if(cseed=0,random(ratio_con_to_zero),\nif(cseed=0.5,random(ratio_con_to_less_than_one),\nif(cseed=1,random(ratio_con_to_one),\n      random(ratio_con_to_big))))\n"}, "percent": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "percent", "definition": "if(seed=-1,random(1..99), if(seed=0,100, random(101..200)))"}, "ratio_con_to_one": {"group": "Ungrouped variables", "templateType": "anything", "description": "<p>These have&nbsp;ratios that converge to 1 and so the ratio test is inconclusive</p>", "name": "ratio_con_to_one", "definition": "if(alt=1,\n[\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*k!/(k+{a})!}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*({a}k+{b})/({d}k^2+{f}k+{g})}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*({a}k^2+{b}k+{c})/({d}k^3+{f}k^2+{g}k+{h})}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*k}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*(k+{a})!/k!}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*({d}k^2+{f}k+{g})/({a}k+{b})}\\$\", \n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*({d}k^3+{f}k^2+{g}k+{h})/({a}k^2+{b}k+{c})}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*{a}*arctan(k)}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*{a}}\\$\",   \n\"\\$\\\\displaystyle\\\\simplify{(-1)^k}\\$\"\n]\n,\n[\n\"\\$\\\\displaystyle\\\\simplify{k!/(k+{a})!}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{({a}k+{b})/({d}k^2+{f}k+{g})}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{({a}k^2+{b}k+{c})/({d}k^3+{f}k^2+{g}k+{h})}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{k}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{(k+{a})!/k!}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{({d}k^2+{f}k+{g})/({a}k+{b})}\\$\", \n\"\\$\\\\displaystyle\\\\simplify{({d}k^3+{f}k^2+{g}k+{h})/({a}k^2+{b}k+{c})}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{*{a}*arctan(k)}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{{a}}\\$\",   \n\"\\$\\\\displaystyle\\\\simplify{1}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{({a}k+{b})/({d}k+{f})}\\$\"\n])"}, "f": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "f", "definition": "coeff[2]"}, "h": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "h", "definition": "coeff[4]"}}, "metadata": {"description": "<p>Test whether a student knows the&nbsp;ratio test of a series, and how to use it. Series include those that the ratio test is inconclusive for. This question could be better in that it could go through the working of determining the limit but I&nbsp;hope to&nbsp;make a separate question which deals with that.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "variablesTest": {"maxRuns": 100, "condition": ""}, "statement": "<p>This question is about the ratio test for series.</p>", "type": "question"}, {"name": "Series: comparison test", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}], "tags": [], "variables": {"pc4": {"group": "Ungrouped variables", "description": "", "definition": "'\\$\\\\displaystyle\\\\simplify{{b}*sin(k)^2/({d}k^{a})}\\$'", "name": "pc4", "templateType": "anything"}, "percent": {"group": "Ungrouped variables", "description": "", "definition": "if(seed=-1,random(1..99), if(seed=0,100, random(101..200)))", "name": "percent", "templateType": "anything"}, "cexpression": {"group": "Ungrouped variables", "description": "", "definition": "switch(cseed='pc1',pc1,cseed='pc2',pc2,cseed='pc3',pc3,cseed='pc4',pc4,cseed='pd1',pd1,cseed='pd2',pd2,cseed='gc1',gc1,cseed='gc2',gc2,cseed='gc3',gc3,pd3)", "name": "cexpression", "templateType": "anything"}, "cseed": {"group": "Ungrouped variables", "description": "<p>pc1 = p series convergent number 1&nbsp;</p>\n<p>gd2= geometric series divergent number 2 etc</p>\n<p></p>", "definition": "if(ccondiv='div',random('pd1','pd2','pd3'),random('pc1','pc2','pc3','pc4','gc1','gc2','gc3'))", "name": "cseed", "templateType": "anything"}, "pc2": {"group": "Ungrouped variables", "description": "", "definition": "'\\$\\\\displaystyle\\\\simplify{{a}/({d}k^{b}-{f}k-{g})}\\$'", "name": "pc2", "templateType": "anything"}, "pd2": {"group": "Ungrouped variables", "description": "", "definition": "'\\$\\\\displaystyle\\\\simplify{{a}/({d}*ln(k))}\\$'", "name": "pd2", "templateType": "anything"}, "a": {"group": "Ungrouped variables", "description": "", "definition": "coeff[0]", "name": "a", "templateType": "anything"}, "gc3": {"group": "Ungrouped variables", "description": "", "definition": "'\\$\\\\displaystyle\\\\simplify{{b}*sin(k)^2/({d}*{a}^k)}\\$'", "name": "gc3", "templateType": "anything"}, "limit": {"group": "Ungrouped variables", "description": "", "definition": "if(seed=1,random(1/9,1/8,1/7,1/5,1/4,1/3,1/2,1/10,1/100),if(seed=0,random(10/9,11/8,8/7,7/5,5/4,4/3,5/2,2,17/10,102/100),1))", "name": "limit", "templateType": "anything"}, "coeff": {"group": "Ungrouped variables", "description": "", "definition": "shuffle(2..12)[0..6]", "name": "coeff", "templateType": "anything"}, "start": {"group": "Ungrouped variables", "description": "", "definition": "coeff[5]", "name": "start", "templateType": "anything"}, "pc1": {"group": "Ungrouped variables", "description": "", "definition": "'\\$\\\\displaystyle\\\\simplify{{a}/({d}k^{b}+{f}k+{g})}\\$'", "name": "pc1", "templateType": "anything"}, "b": {"group": "Ungrouped variables", "description": "", "definition": "coeff[1]", "name": "b", "templateType": "anything"}, "gc1": {"group": "Ungrouped variables", "description": "", "definition": "'\\$\\\\displaystyle\\\\simplify[fractionNumbers]{{a}/({d}^k+{f})}\\$'", "name": "gc1", "templateType": "anything"}, "pd3": {"group": "Ungrouped variables", "description": "", "definition": "'\\$\\\\displaystyle\\\\simplify{{d}k^{b-1}/(sqrt(k^{2*b}+{a}))}\\$'", "name": "pd3", "templateType": "anything"}, "f": {"group": "Ungrouped variables", "description": "", "definition": "coeff[3]", "name": "f", "templateType": "anything"}, "pd1": {"group": "Ungrouped variables", "description": "", "definition": "'\\$\\\\displaystyle\\\\simplify{{a}ln(k)/({d}*k)}\\$'", "name": "pd1", "templateType": "anything"}, "pc3": {"group": "Ungrouped variables", "description": "", "definition": "'\\$\\\\displaystyle\\\\simplify{{b}*cos(k)^2/({d}k^{a})}\\$'", "name": "pc3", "templateType": "anything"}, "seed": {"group": "Ungrouped variables", "description": "", "definition": "random(0..3)", "name": "seed", "templateType": "anything"}, "d": {"group": "Ungrouped variables", "description": "", "definition": "coeff[2]", "name": "d", "templateType": "anything"}, "ccondiv": {"group": "Ungrouped variables", "description": "", "definition": "random('con','div')", "name": "ccondiv", "templateType": "anything"}, "g": {"group": "Ungrouped variables", "description": "", "definition": "coeff[4]", "name": "g", "templateType": "anything"}, "gc2": {"group": "Ungrouped variables", "description": "", "definition": "'\\$\\\\displaystyle\\\\simplify{{b}*cos(k)^2/({d}*{a}^k)}\\$'", "name": "gc2", "templateType": "anything"}}, "rulesets": {}, "variable_groups": [], "functions": {}, "ungrouped_variables": ["seed", "percent", "limit", "ccondiv", "cseed", "cexpression", "coeff", "a", "b", "d", "f", "g", "start", "pc1", "pc2", "pc3", "pc4", "pd1", "pd2", "pd3", "gc1", "gc2", "gc3"], "statement": "<p>This question is about the&nbsp;comparison test for series.</p>", "metadata": {"description": "<p>Test whether a student knows the&nbsp;comparison test of a series, and how to use it. Series include those that the&nbsp;comparison test is inconclusive for.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "parts": [{"displayType": "dropdownlist", "minMarks": 0, "shuffleChoices": false, "prompt": "<p><span data-jme-visible=\"seed=3\">You have a series&nbsp;$\\sum t_k$ with positive terms. Each term in the series is greater than or equal to the corresponding term in another series of positive terms which is actually known to converge.&nbsp;</span></p>\n<p><span data-jme-visible=\"seed=2\">You have a series&nbsp;$\\sum t_k$ with positive terms. Each term in the series is greater than or equal to the corresponding term in another series of positive terms which is actually known to diverge.</span></p>\n<p><span data-jme-visible=\"seed=1\">You have a series&nbsp;$\\sum t_k$ with positive terms. Each term in the series is&nbsp;less than or equal to the corresponding term in another series of positive terms which is actually known to converge.</span></p>\n<p><span data-jme-visible=\"seed=0\">You have a series&nbsp;$\\sum t_k$ with positive terms. Each term in the series is&nbsp;less than or equal to the corresponding term in another series of positive terms which is actually known to diverge.</span></p>\n<p>What does the <strong>comparison test</strong> tell us about the series $\\sum t_k$?</p>", "displayColumns": 0, "variableReplacements": [], "type": "1_n_2", "showFeedbackIcon": true, "scripts": {}, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "marks": 0, "maxMarks": 0, "distractors": ["", "", ""], "choices": ["<p>This series converges.</p>", "<p>This series diverges.</p>", "<p>It doesn't tell us anything.</p>"], "matrix": ["if(seed=1,1,0)", "if(seed=2,1,0)", "if(seed=0 or seed=3,1,0)"]}, {"displayType": "dropdownlist", "minMarks": 0, "shuffleChoices": false, "prompt": "<p><span data-jme-visible=\"seed=0\">Suppose you have two series, $\\sum a_n$ and $\\sum b_n$, where $a,b&gt;0$, $a_n\\ge b_n$ for all $n$, and $\\sum a_n$ is convergent.</span></p>\n<p><span data-jme-visible=\"seed=1\">Suppose you have two series, $\\sum a_n$ and $\\sum b_n$, where $a,b&gt;0$, $a_n\\ge b_n$ for all $n$, and $\\sum a_n$ is divergent.</span></p>\n<p><span data-jme-visible=\"seed=2\">Suppose you have two series, $\\sum a_n$ and $\\sum b_n$, where $a,b&gt;0$, $a_n\\le b_n$ for all $n$, and $\\sum a_n$ is convergent.</span></p>\n<p><span data-jme-visible=\"seed=3\">Suppose you have two series, $\\sum a_n$ and $\\sum b_n$, where $a,b&gt;0$, $a_n\\le b_n$ for all $n$, and $\\sum a_n$ is divergent.</span></p>\n<p>What does the <strong>comparison&nbsp;test</strong> tell us about the series $\\sum b_n$?</p>", "displayColumns": 0, "variableReplacements": [], "type": "1_n_2", "showFeedbackIcon": true, "scripts": {}, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "marks": 0, "maxMarks": 0, "distractors": ["", "", ""], "choices": ["<p>This series converges.</p>", "<p>This series diverges.</p>", "<p>It doesn't tell us anything.</p>"], "matrix": ["if(seed=0,1,0)", "if(seed=3,1,0)", "if(seed=1 or seed=2,1,0)"]}, {"displayType": "dropdownlist", "minMarks": 0, "shuffleChoices": false, "prompt": "<p>Given the series</p>\n<p style=\"text-align: center;\">$\\displaystyle\\sum_{k=\\var{start}}^\\infty$ {cexpression}</p>\n<p>What does the <strong>comparison&nbsp;test</strong> tell us about this series?</p>", "displayColumns": 0, "variableReplacements": [], "type": "1_n_2", "showFeedbackIcon": true, "scripts": {}, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "marks": 0, "maxMarks": 0, "distractors": ["", "", ""], "choices": ["<p>This series converges.&nbsp;</p>", "<p>This series diverges.</p>", "<p>It doesn't tell us anything.</p>"], "matrix": ["if(ccondiv='con',1,0)", "if(ccondiv='div',1,0)", "0"]}], "preamble": {"js": "", "css": ""}, "advice": "<p>The&nbsp;comparison test is as follows. Suppose that $0\\ge a_n\\ge b_n$ for sufficiently large $n$.</p>\n<p>&bull; If $a_n$ diverges, then $b_n$ also diverges.</p>\n<p>&bull; If $b_n$ converges, then $a_n$ also converges.</p>\n<p>Notice, if $a_n$ converges or if $b_n$ diverges the test doesn't say anything, in these&nbsp;cases some might say the test is 'inconclusive' or 'fails'.&nbsp;</p>\n<p></p>\n<p>a)</p>\n<p><span data-jme-visible=\"seed=1\">The comparison test ensures us that our series converges (because a larger one does).</span></p>\n<p><span data-jme-visible=\"seed=2\">The comparison test ensures us that our series diverges (because a smaller one does).</span></p>\n<p><span data-jme-visible=\"seed=0 or seed=3\">The comparison test doesn't tell us anything about this situation.</span></p>\n<p></p>\n<p>b)</p>\n<p><span data-jme-visible=\"seed=0\">The comparison test ensures us that our series converges (because a larger one does).</span></p>\n<p><span data-jme-visible=\"seed=3\">The comparison test ensures us that our series diverges (because a smaller one does).</span></p>\n<p><span data-jme-visible=\"seed=1 or seed=2\">The comparison test doesn't tell us anything about this situation.</span></p>\n<p></p>\n<p>c)</p>\n<p><span data-jme-visible=\"cseed='pc1'\">In the denominator of {cexpression} the dominant term is $\\var{d}k^\\var{b}$,&nbsp;so we will compare our series with $\\sum_{k=\\var{start}}^\\infty\\frac{\\var{a}}{\\var{d}k^\\var{b}}$ which is a $p$-series with $p=\\var{b}$ and hence is convergent.&nbsp;Now for $k\\ge\\var{start}$ we definitely have</span></p>\n<p><span data-jme-visible=\"cseed='pc1'\">\\[\\simplify{{a}/({d}k^{b}+{f}k+{g})&lt;{a}/({d}k^{b})}\\]</span></p>\n<p><span data-jme-visible=\"cseed='pc1'\">and therefore our series is also convergent.</span></p>\n<p><span data-jme-visible=\"cseed='pc2'\">In the denominator of {cexpression} the dominant term is $\\var{d}k^\\var{b}$,&nbsp;so we will compare&nbsp;our series with $\\sum_{k=\\var{start}}^\\infty\\frac{\\var{a}}{\\var{d}k^\\var{b}}$ which is a $p$-series with $p=\\var{b}$ and hence is convergent. However, for $k\\ge\\var{start}$ we have</span></p>\n<p><span data-jme-visible=\"cseed='pc2'\">\\[\\simplify{{a}/({d}k^{b}-{f}k-{g})&gt;{a}/({d}k^{b})}\\]</span></p>\n<p><span data-jme-visible=\"cseed='pc2'\">and so can't use the comparison test to compare it to that series. But, for $k\\ge\\var{start}$ we definitely have</span></p>\n<p><span data-jme-visible=\"cseed='pc2'\">\\[\\simplify{{a}/({d}k^{b}-{f}k-{g})&lt;{a}/({d}k^{b+1})}\\]</span></p>\n<p><span data-jme-visible=\"cseed='pc2'\">and $\\sum_{k=\\var{start}}^\\infty \\simplify{{a}/({d}k^{b+1})}$ is a convergent $p$-series. Therefore our series is also convergent.</span></p>\n<p><span data-jme-visible=\"cseed='pc3'\">Notice that $0\\le \\cos^2(k)\\le 1$ and so for $k\\ge\\var{start}$ we definitely have&nbsp;</span></p>\n<p><span data-jme-visible=\"cseed='pc3'\">\\[\\simplify{{b}cos(k)^2/({d}k^{a})&lt;={b}/({d}k^{a})}\\]</span></p>\n<p><span data-jme-visible=\"cseed='pc3'\">Also notice, $\\sum_{k=\\var{start}}^\\infty \\frac{\\var{b}}{\\var{d}k^{\\var{a}}}$ is a convergent&nbsp;$p$-series with $p=\\var{a}$. Therefore our series is also convergent.</span></p>\n<p><span data-jme-visible=\"cseed='pc4'\">Notice that $0\\le \\sin^2(k)\\le 1$ and so for $k\\ge\\var{start}$ we definitely have&nbsp;</span></p>\n<p><span data-jme-visible=\"cseed='pc4'\">\\[\\simplify{{b}sin(k)^2/({d}k^{a})&lt;={b}/({d}k^{a})}\\]</span></p>\n<p><span data-jme-visible=\"cseed='pc4'\">Also notice, $\\sum_{k=\\var{start}}^\\infty \\frac{\\var{b}}{\\var{d}k^{\\var{a}}}$ is a convergent $p$-series with $p=\\var{a}$. Therefore our series is also convergent.</span></p>\n<p><span data-jme-visible=\"cseed='pd1'\">For $k\\ge\\var{start}$ we definitely have</span></p>\n<p><span data-jme-visible=\"cseed='pd1'\">\\[\\simplify{{a}ln(k)/({d}k)&gt;{a}/({d}k)}\\]</span></p>\n<p><span data-jme-visible=\"cseed='pd1'\">and $\\sum_{k=\\var{start}}^\\infty \\simplify{{a}/({d}k)}$ is a divergent series (it is a scalar multiple of the harmonic series or a $p$-series with $p=1$). Therefore our series is also divergent.</span></p>\n<p><span data-jme-visible=\"cseed='pd2'\">For $k\\ge\\var{start}$ we definitely have</span></p>\n<p><span data-jme-visible=\"cseed='pd2'\">\\[\\simplify{{a}/({d}ln(k))&gt;{a}/({d}k)}\\]</span></p>\n<p><span data-jme-visible=\"cseed='pd2'\">and $\\sum_{k=\\var{start}}^\\infty \\simplify{{a}/({d}k)}$ is a divergent series (it is a scalar multiple of the harmonic series or a $p$-series with $p=1$). Therefore our series is also divergent.</span></p>\n<p><span data-jme-visible=\"cseed='pd3'\">Given $\\simplify{{d}k^{b-1}/(sqrt(k^{2*b}+{a}))}$ we might&nbsp;realise that&nbsp;</span></p>\n<p><span data-jme-visible=\"cseed='pd3'\">\\[\\simplify{{d}k^{b-1}/(sqrt(k^{2*b}+{a}))&lt;{d}k^{b-1}/(sqrt(k^{2*b}))={d}k^{b-1}/(k^{b})={d}/k}\\]</span></p>\n<p><span data-jme-visible=\"cseed='pd3'\">however, $\\sum \\simplify{{d}/k}$ is divergent (it is a scalar multiple of the harmonic series or a $p$-series with $p=1$) and given the direction of the inequality we can't use the comparison test to test these two series. But, lets try something else:</span></p>\n<p><span data-jme-visible=\"cseed='pd3'\">\\[\\simplify{{d}k^{b-1}/(sqrt(k^{2*b}+{a}))&gt;{d}k^{b-1}/(sqrt(k^{2*b}+{a}k^{2*b}))={d}k^{b-1}/(sqrt({a+1}k^{2*b}))={d}k^{b-1}/(sqrt({a+1})k^{b})={d}/(sqrt({a+1})k)}\\]</span></p>\n<p><span data-jme-visible=\"cseed='pd3'\">Notice, $\\sum_{k=\\var{start}}^\\infty \\simplify{{d}/(sqrt({a+1})k)}$ is a divergent series (it is a scalar multiple of the harmonic series or a $p$-series with $p=1$) and so our series is too.</span></p>\n<p><span data-jme-visible=\"cseed='gc1'\">In the denominator of {cexpression} the dominant term is $\\var{d}^k$,&nbsp;so we will compare&nbsp;our series with $\\sum_{k=\\var{start}}^\\infty\\frac{\\var{a}}{\\var{d}^k}$ which is the same as $\\sum_{k=\\var{start}}^\\infty\\var{a}\\left(\\frac{1}{\\var{d}}\\right)^k$ and so is a convergent geometric series with common ratio $r=\\frac{1}{\\var{d}}$.&nbsp;Now for $k\\ge\\var{start}$ we definitely have</span></p>\n<p><span data-jme-visible=\"cseed='gc1'\">\\[\\simplify{{a}/({d}^k+{f})&lt;{a}/({d}^k)}\\]</span></p>\n<p><span data-jme-visible=\"cseed='gc1'\">and therefore our series is also convergent.</span></p>\n<p><span data-jme-visible=\"cseed='gc2'\">Notice that $0\\le \\cos^2(k)\\le 1$ and so for $k\\ge\\var{start}$ we definitely have&nbsp;</span></p>\n<p><span data-jme-visible=\"cseed='gc2'\">\\[\\simplify{{b}cos(k)^2/({d}{a}^k)&lt;={b}/({d}{a}^k)}\\]</span></p>\n<p><span data-jme-visible=\"cseed='gc2'\">Also notice, $\\sum_{k=\\var{start}}^\\infty \\frac{\\var{b}}{\\var{d}\\times\\var{a}^k}$ is a convergent&nbsp;$p$-series with&nbsp;with $p=\\var{a}$. Therefore our series is also convergent.</span></p>\n<p><span data-jme-visible=\"cseed='gc3'\">Notice that $0\\le \\sin^2(k)\\le 1$ and so for $k\\ge\\var{start}$ we definitely have&nbsp;</span></p>\n<p><span data-jme-visible=\"cseed='gc3'\">\\[\\simplify{{b}sin(k)^2/({d}*{a}^k)&lt;={b}/({d}*{a}^k)}\\]</span></p>\n<p><span data-jme-visible=\"cseed='gc3'\">Also notice, $\\sum_{k=\\var{start}}^\\infty \\frac{\\var{b}}{\\var{d}\\times\\var{a}^k}$ is a convergent&nbsp;$p$-series with&nbsp;with $p=\\var{a}$. Therefore our series is also convergent.</span></p>\n<p></p>", "variablesTest": {"condition": "", "maxRuns": 100}, "type": "question"}]}], "feedback": {"showanswerstate": true, "advicethreshold": 0, "intro": "", "showtotalmark": true, "feedbackmessages": [], "allowrevealanswer": true, "showactualmark": true, "enterreviewmodeimmediately": true, "showexpectedanswerswhen": "inreview", "showpartfeedbackmessageswhen": "always", "showactualmarkwhen": "always", "showtotalmarkwhen": "always", "showanswerstatewhen": "always", "showadvicewhen": "never"}, "showstudentname": true, "timing": {"timedwarning": {"action": "none", "message": ""}, "timeout": {"action": "none", "message": ""}, "allowPause": true}, "metadata": {"licence": "None specified", "description": ""}, "name": "analysis", "percentPass": "40", "navigation": {"showresultspage": "oncompletion", "allowregen": true, "browse": true, "reverse": true, "startpassword": "", "onleave": {"action": "none", "message": ""}, "preventleave": true, "showfrontpage": true}, "duration": 0, "type": "exam", "contributors": [{"name": "Daniel Atcheson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3379/"}], "extensions": ["jsxgraph"], "custom_part_types": [], "resources": []}