[[0]]
\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": ["{Ch1}
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\"", "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": "\"If a function $f: \\\\mathbb{R} \\\\to \\\\mathbb{R}$ is continuous on $(a,b)$ and $f(a) < \\\\gamma < f(b)$, then $f(c)=\\\\gamma$ for some $c \\\\in (a,b)$.
\"", "name": "f2", "description": ""}, "h": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "name": "h", "description": ""}, "tr1": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If a function $f: \\\\mathbb{R} \\\\to \\\\mathbb{R}$ is differentiable at $c \\\\in \\\\mathbb{R}$, then it is continuous at $c$.
\"", "name": "tr1", "description": ""}, "f4": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If a function $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then $f\\'(c)=0$ for some $c \\\\in (a,b)$.
\"", "name": "f4", "description": ""}, "f3": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"Given any function defined on $[a,b]$ with $f(a) < \\\\gamma < f(b)$, then $f(c)=\\\\gamma$ for some $c \\\\in (a,b)$.
\"", "name": "f3", "description": ""}, "tr5": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If a function $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then $f\\'(c)=\\\\dfrac{f(b)-f(a)}{b-a}$ for some $c \\\\in (a,b)$.
\"", "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": "\"If a function $f$ is differentiable on $(a,b)$, then $f\\'(c)=\\\\dfrac{f(b)-f(a)}{b-a}$ for some $c \\\\in (a,b)$.
\"", "name": "f5", "description": ""}, "tr2": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If a function $f$ is continuous on $[a,b]$ and $f(a) < \\\\gamma < f(b)$, then $f(c)=\\\\gamma$ for some $c \\\\in (a,b)$.
\"", "name": "tr2", "description": ""}, "f": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..3)", "name": "f", "description": ""}, "tr3": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If a function $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, and if $f(a) < \\\\gamma < f(b)$, then $f(c)=\\\\gamma$ for some $c \\\\in (a,b)$.
\"", "name": "tr3", "description": ""}, "f6": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If a function $f$ is continuous and differentiable on $(a,b)$, then $f\\'(c)=\\\\dfrac{f(b)-f(a)}{b-a}$ for some $c \\\\in (a,b)$.
\"", "name": "f6", "description": ""}, "tr6": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If a function $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, and if $f\\'(x) >0$ for all $x \\\\in (a,b)$, then $f(b)>f(a)$.
\"", "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": "\"If a function $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, and if $f(a) = f(b)$, then $f\\'(c)=0$ for some $c \\\\in (a,b)$.
\"", "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": "Answer the following question on continuity and differentiability. Note that every correct answer is worth 1 mark, but every wrong answer loses a mark.
", "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": "Multiple response question (2 correct out of 4) covering properties of continuity and differentiability. Selection of questions from a pool.
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\"", "name": "tr8", "description": ""}, "x": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "name": "x", "description": ""}, "f1": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If $a_n \\\\geq \\\\dfrac{1}{n^2}$ for all $n \\\\in \\\\mathbb{N}$, then $\\\\Sigma a_n$ converges.
\"", "name": "f1", "description": ""}, "ch2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(u=1,tr5,if(u=2,tr6,if(u=3,tr7,tr8)))", "name": "ch2", "description": ""}, "f20": {"templateType": "anything", "group": "Ungrouped variables", "definition": "'It is not possible for an unbounded sequence to have a bounded subsequence.'", "name": "f20", "description": ""}, "tr13": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If a power series $\\\\Sigma a_n x^n$ has radius of convergence $R$ with $0 < R <\\\\infty$, then the power series converges for all $x$ with $|x|<R$.
\"", "name": "tr13", "description": ""}, "tr3": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If $a_n \\\\not\\\\to 0$ as $n \\\\to \\\\infty$, then the series $\\\\Sigma a_n$ diverges.
\"", "name": "tr3", "description": ""}, "f4": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If the series $\\\\Sigma a_n$ diverges, then $a_n \\\\not\\\\to 0$ as $n \\\\to \\\\infty$.
\"", "name": "f4", "description": ""}, "f3": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If $a_n \\\\to 0$ as $n \\\\to \\\\infty$, then the series $\\\\Sigma a_n$ converges.
\"", "name": "f3", "description": ""}, "tr2": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If $a_n \\\\geq \\\\dfrac{1}{n}$ for all $n \\\\in \\\\mathbb{N}$, then $\\\\Sigma a_n$ diverges.
\"", "name": "tr2", "description": ""}, "f15": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If a power series $\\\\Sigma a_n x^n$ has radius of convergence $R$ with $0<R<\\\\infty$, then $a_n \\\\neq 0$ and $\\\\dfrac{a_{n+1}}{a_n}$ tends to a limit as $n \\\\to \\\\infty$.
\"", "name": "f15", "description": ""}, "f9": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If $\\\\Sigma a_n$ is convergent then it is absolutely convergent.
\"", "name": "f9", "description": ""}, "tr11": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If $a_n = (-1)^{n-1} u_n$ with $u_n >0$ and if $\\{u_n\\}$ does not converge to $0$, then $\\\\Sigma a_n$ diverges.
\"", "name": "tr11", "description": ""}, "f10": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If $\\\\Sigma a_n$ is not divergent then it is absolutely convergent.
\"", "name": "f10", "description": ""}, "tr6": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If $a_n>0$ for all $n \\\\in \\\\mathbb{N}$ and if $\\\\dfrac{a_{n+1}}{a_n} \\\\to \\\\ell >1$ as $n \\\\to \\\\infty$, then $\\\\Sigma a_n$ diverges.
\"", "name": "tr6", "description": ""}, "tr16": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If a power series $\\\\Sigma a_n x^n$ diverges for some value $x=X \\\\neq 0$, then the radius of convergence $R$ satisfies $R \\\\leq |X|$.
\"", "name": "tr16", "description": ""}, "ch8": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(x=1,f13,if(x=2,f14,if(f=x,f15,f16)))", "name": "ch8", "description": ""}, "f13": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If a power series $\\\\Sigma a_n x^n$ has radius of convergence $R$ with $0 < R <\\\\infty$, then the power series converges for $x=R$.
\"", "name": "f13", "description": ""}, "f14": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If a power series $\\\\Sigma a_n x^n$ has radius of convergence $R$ with $0 < R <\\\\infty$, then the power series diverges for $x=R$.
\"", "name": "f14", "description": ""}, "tr1": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If $a_n \\\\geq 0$ for all $n \\\\in \\\\mathbb{N}$ and if $a_n \\\\leq \\\\dfrac{1}{n^2}$ for all $n \\\\in \\\\mathbb{N}$, then $\\\\Sigma a_n$ converges.
\"", "name": "tr1", "description": ""}, "g": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "name": "g", "description": ""}, "u": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "name": "u", "description": ""}, "w": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "name": "w", "description": ""}, "f11": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If $a_n = (-1)^{n-1} u_n$ with $u_n >0$ and if $\\{u_n\\}$ is not decreasing, then $\\\\Sigma a_n$ diverges.
\"", "name": "f11", "description": ""}, "v": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "name": "v", "description": ""}, "tr14": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If a power series $\\\\Sigma a_n x^n$ has radius of convergence $R$ with $0 < R <\\\\infty$, then the power series diverges for all $x$ with $|x|>R$.
\"", "name": "tr14", "description": ""}, "ch1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(t=1,tr1,if(t=2,tr2,if(t=3,tr3,tr4)))", "name": "ch1", "description": ""}, "tr10": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If $\\\\Sigma a_n$ is divergent then it is not absolutely convergent.
\"", "name": "tr10", "description": ""}, "t": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "name": "t", "description": ""}, "tr12": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If $a_n = (-1)^{n-1} u_n$ with $u_n >0$ and if $\\{u_n\\}$ is increasing, then $\\\\Sigma a_n$ diverges.
\"", "name": "tr12", "description": ""}, "tr20": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If $\\\\Sigma a_n$ is divergent then it is not absolutely convergent.
\"", "name": "tr20", "description": ""}, "f7": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If $a_n > 0$ for all $n \\\\in \\\\mathbb{N}$ and if $\\\\dfrac{a_{n+1}}{a_n}$ $\\\\to$ $\\\\ell >0$ as $n \\\\to \\\\infty$, then $\\\\Sigma a_n$ converges.
\"", "name": "f7", "description": ""}, "tr7": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If $a_n \\\\neq 0$ for all $n \\\\in \\\\mathbb{N}$ and if $\\\\dfrac{a_{n+1}}{a_n}$ $\\\\to$ $\\\\infty$ as $n \\\\to \\\\infty$, then $\\\\Sigma a_n$ diverges.
\"", "name": "tr7", "description": ""}, "tr9": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If $\\\\Sigma a_n$ is absolutely convergent then it is convergent.
\"", "name": "tr9", "description": ""}, "h": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "name": "h", "description": ""}, "f2": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If $a_n \\\\geq 0$ for all $n \\\\in \\\\mathbb{N}$ and if $a_n \\\\leq \\\\dfrac{1}{n}$ for all $n \\\\in \\\\mathbb{N}$, then $\\\\Sigma a_n$ converges.
\"", "name": "f2", "description": ""}, "tr15": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If a power series $\\\\Sigma a_n x^n$ converges for some value $x=X \\\\neq 0$, then the radius of convergence $R$ satisfies $R \\\\geq |X|$.
\"", "name": "tr15", "description": ""}, "ch6": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(g=1,f5,if(g=6,f2,if(g=3,f7,f8)))", "name": "ch6", "description": ""}, "tr5": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If $a_n > 0$ for all $n \\\\in \\\\mathbb{N}$ and if $\\\\dfrac{a_{n+1}}{a_n} \\\\to \\\\ell <1$ as $n \\\\to \\\\infty$, then $\\\\Sigma a_n$ converges.
\"", "name": "tr5", "description": ""}, "f5": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If $a_n > 0$ for all $n \\\\in \\\\mathbb{N}$ and if $\\\\dfrac{a_{n+1}}{a_n} \\\\to \\\\ell =1$ as $n \\\\to \\\\infty$, then $\\\\Sigma a_n$ converges.
\"", "name": "f5", "description": ""}, "f": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "name": "f", "description": ""}, "ch7": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(h=1,f9,if(h=2,f10,if(h=3,f11,f12)))", "name": "ch7", "description": ""}, "f8": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If $a_n \\\\neq 0$ for all $n \\\\in \\\\mathbb{N}$ and if $\\\\dfrac{a_{n+1}}{a_n}$ $\\\\to$ $\\\\ell<1$ as $n \\\\to \\\\infty$, then $\\\\Sigma a_n$ converges.
\"", "name": "f8", "description": ""}, "f12": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If $a_n = (-1)^{n-1} u_n$ with $u_n \\\\geq 0$ and if $\\{u_n\\}$ is increasing, then $\\\\Sigma a_n$ diverges.
\"", "name": "f12", "description": ""}, "f6": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If $a_n>0$ for all $n \\\\in \\\\mathbb{N}$ and if $\\\\dfrac{a_{n+1}}{a_n}$ $\\\\to$ $\\\\ell=1$ as $n \\\\to \\\\infty$, then $\\\\Sigma a_n$ diverges.
\"", "name": "f6", "description": ""}, "ch3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(v=1,tr9,if(v=2,tr10,if(v=3,tr11,tr12)))", "name": "ch3", "description": ""}, "ch4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(w=1,tr13,if(w=2,tr14,if(w=3,tr15,tr16)))", "name": "ch4", "description": ""}, "tr4": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If the series $\\\\Sigma a_n$ converges, then $a_n \\\\to 0$ as $n \\\\to \\\\infty$.
\"", "name": "tr4", "description": ""}, "ch5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(f=1,f1,if(f=2,f2,if(f=3,f3,f4)))", "name": "ch5", "description": ""}, "f16": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If a power series $\\\\Sigma a_n x^n$ with $a_n \\\\neq 0$ for all $n$ has radius of convergence $R$ with $R<\\\\infty$, then $\\\\dfrac{a_{n+1}}{a_n}$ tends to a limit as $n \\\\to \\\\infty$.
\"", "name": "f16", "description": ""}}, "ungrouped_variables": ["f1", "f2", "f3", "f4", "f5", "f6", "f7", "f8", "f9", "f10", "f11", "f12", "f13", "f14", "f15", "f16", "tr1", "tr2", "tr3", "tr4", "tr5", "tr6", "tr7", "tr8", "tr9", "tr10", "tr11", "tr12", "tr13", "tr14", "tr15", "tr16", "t", "u", "v", "w", "f", "g", "h", "x", "ch1", "ch2", "ch3", "ch4", "ch5", "ch6", "ch7", "ch8", "f20", "tr20"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"displayType": "radiogroup", "layout": {"expression": ""}, "choices": ["{Ch1}
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\n \n \n \n", "showCorrectAnswer": true, "marks": 0}], "statement": "Answer the following question on series. Note that every correct answer is worth 1 mark, but every wrong answer loses a mark.
", "tags": ["checked2015", "divergent series", "limits", "MAS1601", "MAS2224", "power series", "series"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "17/04/2015:
\n(OK) new question based on a similar style question on sequences. Changed the statements to long text to enable better mathematical expressions. Encountered problems when editing (math expressions not recognised).
", "licence": "Creative Commons Attribution 4.0 International", "description": "Multiple response question (4 correct out of 8) covering properties of convergent and divergent series and including questions on power series. Selection of questions from a pool.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "You should be able to work out the correct answers from your notes.
"}, {"name": "True/false statements about limits of sequences", "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": {"tr8": {"group": "Ungrouped variables", "templateType": "long string", "definition": "\"A divergent sequence can have a convergent subsequence.
\"", "name": "tr8", "description": ""}, "x": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..4)", "name": "x", "description": ""}, "f1": {"group": "Ungrouped variables", "templateType": "long string", "definition": "\"A bounded sequence is convergent.
\"", "name": "f1", "description": ""}, "ch2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(u=1,tr4,if(u=2,tr5,tr6))", "name": "ch2", "description": ""}, "f20": {"group": "Ungrouped variables", "templateType": "anything", "definition": "'It is not possible for an unbounded sequence to have a bounded subsequence.'", "name": "f20", "description": ""}, "f4": {"group": "Ungrouped variables", "templateType": "long string", "definition": "\"If {$\\{x_n\\}$} converges, then {$\\{x_{n+i}\\}$} could diverge for some natural number $i$.
\"", "name": "f4", "description": ""}, "f3": {"group": "Ungrouped variables", "templateType": "long string", "definition": "\"A convergent sequence is either increasing or decreasing.
\"", "name": "f3", "description": ""}, "tr2": {"group": "Ungrouped variables", "templateType": "long string", "definition": "\"If {$\\{x_{n+i}\\}$} diverges for some natural number $i$, then {$\\{x_n\\}$} diverges.
\"", "name": "tr2", "description": ""}, "f9": {"group": "Ungrouped variables", "templateType": "anything", "definition": "'There exists a sequence that is not bounded but which converges.'", "name": "f9", "description": "There exists a sequence that is not bounded but which converges.
"}, "tr11": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random('A finite set is bounded.', 'A set that is not bounded has an infinite number of elements.')", "name": "tr11", "description": ""}, "tr3": {"group": "Ungrouped variables", "templateType": "long string", "definition": "\"If a sequence is not bounded, then it does not converge.
\"", "name": "tr3", "description": ""}, "tr6": {"group": "Ungrouped variables", "templateType": "anything", "definition": "'A sequence with only a finite number of non zero terms converges to 0.'", "name": "tr6", "description": ""}, "ch8": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(x=1,f11,if(x=2,f12,if(x=3,f13,f14)))", "name": "ch8", "description": ""}, "f13": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random('A bounded set never has a maximum element.', 'A bounded set never has a minimum element.')", "name": "f13", "description": ""}, "f14": {"group": "Ungrouped variables", "templateType": "anything", "definition": "'A set with an infinite number of elements cannot be bounded.'", "name": "f14", "description": ""}, "tr1": {"group": "Ungrouped variables", "templateType": "long string", "definition": "\"A convergent sequence is bounded.
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", "tags": ["bounded sequences", "bounded sets", "checked2015", "convergence", "convergent sequences", "decreasing sequence", "divergence", "divergent sequences", "increasing sequence", "limits", "monotone sequence", "monotonic sequence", "sequence", "sequences", "subsequence", "tested1", "unbounded sequences"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "You should be able to work out the correct answers from your notes.
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\"", "description": "", "name": "tr4"}, "tr1": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If $f(x) \\\\to \\\\ell$ as $x \\\\to c$ and if $x_n \\\\to c$ as $n \\\\to \\\\infty$ (with each $x_n \\\\neq c$), then $f(x_n) \\\\to \\\\ell$ as $n \\\\to \\\\infty$.
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", "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": "Multiple response question (2 correct out of 4) covering properties of continuity and limits of functions. Selection of questions from a pool.
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\"", "description": "", "name": "f1"}, "ch2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(u=1,tr4,if(u=2,tr5,if(u=3,tr6, if(u=4, tr7,tr8))))", "description": "", "name": "ch2"}, "u": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "", "name": "u"}, "f7": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If a bounded function $f$ is Riemann integrable on $[a,b]$, and if $P_n$ ($n \\\\in \\\\mathbb{N}$) are partitions of $[a,b]$, then $U(P_n)-L(P_n) \\\\to 0$ as $n \\\\to \\\\infty$.
\"", "description": "", "name": "f7"}, "tr7": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If $f$ is a bounded function on $[a,b]$, and if $P_n$ ($n \\\\in \\\\mathbb{N}$) are partitions of $[a,b]$ such that $U(P_n)-L(P_n) \\\\to 0$ as $n \\\\to \\\\infty$, then $f$ is Riemann integrable on $[a,b]$.
\"", "description": "", "name": "tr7"}, "f5": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If $f$ is a bounded function on $[a,b]$, and if $P$ and $Q$ are partitions of $[a,b]$ with $Q$ a refinement of $P$, then $L(P) \\\\geq U(Q)$.
\"", "description": "", "name": "f5"}, "h": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "description": "", "name": "h"}, "f4": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If $f$ is a bounded function on $[a,b]$, and if $P$ and $Q$ are partitions of $[a,b]$, then $L(P) \\\\leq L(Q)$.
\"", "description": "", "name": "f4"}, "f3": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If a function $f$ is Riemann integrable on $[a,b]$, then $f$ is bounded and increasing on $[a,b]$.
\"", "description": "", "name": "f3"}, "tr5": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If $f$ is a bounded function on $[a,b]$, and if $P$ and $Q$ are partitions of $[a,b]$ with $Q$ a refinement of $P$, then $L(P) \\\\leq L(Q)$.
\"", "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": "\"If $f$ is a bounded function on $[a,b]$, and if $P$ and $Q$ are partitions of $[a,b]$, then $U(P \\\\cup Q) \\\\leq U(P)$.
\"", "description": "", "name": "tr6"}, "tr2": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If a function $f$ is bounded and decreasing on $[a,b]$, then $f$ is Riemann integrable on $[a,b]$.
\"", "description": "", "name": "tr2"}, "f2": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If a function $f$ is Riemann integrable on $[a,b]$, then $f$ is bounded and decreasing on $[a,b]$.
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\"", "description": "", "name": "tr3"}, "f8": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If a function $f$ is Riemann integrable on $[a,b]$, then $f$ is continuous on $[a,b]$.
\"", "description": "", "name": "f8"}, "f6": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If $f$ is a bounded function on $[a,b]$, and if $P_n$ ($n \\\\in \\\\mathbb{N}$) are partitions of $[a,b]$, then $U(P_n)-L(P_n) \\\\to 0$ as $n \\\\to \\\\infty$.
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\n(OK) new question adapting the format of an older question
", "licence": "Creative Commons Attribution 4.0 International", "description": "Multiple response question (2 correct out of 4) covering properties of Riemann integration. Selection of questions from a pool.
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