// Numbas version: exam_results_page_options
{"name": "James's copy of When does a sequence get within $d$ of its limit?", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"preamble": {"css": "", "js": ""}, "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>", "ungrouped_variables": ["a", "c", "b", "s1", "b1", "d", "r", "n", "tval", "mono", "t1", "t2", "t3", "u1", "u2", "v1", "v2", "w1"], "name": "James's copy of When does a sequence get within $d$ of its limit?", "functions": {"chcop": {"definition": "if(gcd(a,b)=1,b,chcop(a,random(1..20)))", "language": "jme", "type": "number", "parameters": [["a", "number"], ["b", "number"]]}}, "tags": ["checked2015", "convergence of a sequence", "limit", "limit of a sequence", "Limits", "limits", "query", "sequences", "taking the limit", "tested1", "udf"], "variablesTest": {"maxRuns": 100, "condition": ""}, "variable_groups": [], "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>"}, "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>", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"scripts": {}, "type": "gapfill", "sortAnswers": false, "showFeedbackIcon": true, "prompt": "<p>Find the limit $\\ell$ of $\\{x_n\\}$. Input as a fraction or an integer.</p>\n<p>Limit $\\ell=$ [[0]]</p>", "marks": 0, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "unitTests": [], "customMarkingAlgorithm": "", "gaps": [{"scripts": {}, "marks": 2, "type": "jme", "unitTests": [], "showFeedbackIcon": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "showPreview": true, "vsetRange": [0, 1], "answer": "{a}/{c}", "checkVariableNames": false, "failureRate": 1, "customMarkingAlgorithm": "", "vsetRangePoints": 5, "expectedVariableNames": [], "variableReplacementStrategy": "originalfirst", "variableReplacements": []}], "variableReplacementStrategy": "originalfirst", "variableReplacements": []}, {"scripts": {}, "type": "gapfill", "sortAnswers": false, "showFeedbackIcon": 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>", "marks": 0, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "unitTests": [], "customMarkingAlgorithm": "", "gaps": [{"allowFractions": false, "scripts": {}, "mustBeReduced": false, "type": "numberentry", "unitTests": [], "showFeedbackIcon": true, "mustBeReducedPC": 0, "marks": "8", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "notationStyles": ["plain", "en", "si-en"], "maxValue": "{n}", "correctAnswerFraction": false, "customMarkingAlgorithm": "", "minValue": "{n}", "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "variableReplacements": []}], "variableReplacementStrategy": "originalfirst", "variableReplacements": []}, {"maxMarks": 0, "scripts": {}, "minMarks": 0, "type": "1_n_2", "displayType": "radiogroup", "showFeedbackIcon": true, "prompt": "<p>Which one of the following describes $\\{x_n\\}$?</p>", "matrix": ["2", 0, 0], "showCorrectAnswer": true, "marks": 0, "extendBaseMarkingAlgorithm": true, "distractors": ["", "", ""], "choices": ["<p>{u1}</p>", "<p>{u2}</p>", "<p>{t3}</p>"], "shuffleChoices": true, "unitTests": [], "customMarkingAlgorithm": "", "displayColumns": "3", "variableReplacementStrategy": "originalfirst", "variableReplacements": []}], "variables": {"w1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(u1=t1,v1,v2)", "name": "w1", "description": ""}, "t3": {"templateType": "string", "group": "Ungrouped variables", "definition": "\"Neither\"", "name": "t3", "description": ""}, "t1": {"templateType": "string", "group": "Ungrouped variables", "definition": "\"Increasing\"", "name": "t1", "description": ""}, "tval": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sqrt((1 / c) * ((10 ^ r * abs(b * c -(a * d))) / c -d))", "name": "tval", "description": ""}, "u1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(a/c > b/d,t1,t2)", "name": "u1", "description": "<p>Correct answer</p>"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..20)", "name": "a", "description": ""}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(b1*c=a*d,b1+1,b1)", "name": "b", "description": ""}, "v1": {"templateType": "string", "group": "Ungrouped variables", "definition": "\"increasing\"", "name": "v1", "description": ""}, "mono": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(a/c>b/d,1,2)", "name": "mono", "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>"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "chcop(a,a)", "name": "c", "description": ""}, "t2": {"templateType": "string", "group": "Ungrouped variables", "definition": "\"Decreasing\"", "name": "t2", "description": ""}, "u2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(a/c > b/d,t2,t1)", "name": "u2", "description": "<p>Incorrect answer</p>"}, "v2": {"templateType": "string", "group": "Ungrouped variables", "definition": "\"decreasing\"", "name": "v2", "description": ""}, "s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "s1", "description": ""}, "b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(2..9)", "name": "b1", "description": ""}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "chcop(c,c)", "name": "d", "description": ""}, "n": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(isint(tval), tval +1,ceil(tval))", "name": "n", "description": ""}, "r": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2,3,4,5,6)", "name": "r", "description": ""}}, "extensions": [], "type": "question", "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "James Denholm-Price", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/17/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "James Denholm-Price", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/17/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}