You should be able to work out the correct answers from your notes.
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\n \n \n \n", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"maxAnswers": 0, "shuffleChoices": true, "matrix": [[1, -1], [1, -1], [1, -1], ["1", "-1"], [-1, 1], [-1, 1], ["-1", "1"], ["-1", "1"]], "shuffleAnswers": false, "minAnswers": 0, "variableReplacements": [], "answers": [true, false], "warningType": "none", "variableReplacementStrategy": "originalfirst", "maxMarks": 0, "showCorrectAnswer": true, "scripts": {}, "marks": 0, "choices": ["{Ch1}
", "{Ch2}
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", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"f1": {"definition": "\"If $a_n \\\\geq \\\\dfrac{1}{n^2}$ for all $n \\\\in \\\\mathbb{N}$, then $\\\\Sigma a_n$ converges.
\"", "templateType": "long string", "group": "Ungrouped variables", "name": "f1", "description": ""}, "f2": {"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.
\"", "templateType": "long string", "group": "Ungrouped variables", "name": "f2", "description": ""}, "f3": {"definition": "\"If $a_n \\\\to 0$ as $n \\\\to \\\\infty$, then the series $\\\\Sigma a_n$ converges.
\"", "templateType": "long string", "group": "Ungrouped variables", "name": "f3", "description": ""}, "f4": {"definition": "\"If the series $\\\\Sigma a_n$ diverges, then $a_n \\\\not\\\\to 0$ as $n \\\\to \\\\infty$.
\"", "templateType": "long string", "group": "Ungrouped variables", "name": "f4", "description": ""}, "f5": {"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.
\"", "templateType": "long string", "group": "Ungrouped variables", "name": "f5", "description": ""}, "f6": {"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.
\"", "templateType": "long string", "group": "Ungrouped variables", "name": "f6", "description": ""}, "f7": {"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.
\"", "templateType": "long string", "group": "Ungrouped variables", "name": "f7", "description": ""}, "f8": {"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.
\"", "templateType": "long string", "group": "Ungrouped variables", "name": "f8", "description": ""}, "f9": {"definition": "\"If $\\\\Sigma a_n$ is convergent then it is absolutely convergent.
\"", "templateType": "long string", "group": "Ungrouped variables", "name": "f9", "description": ""}, "tr20": {"definition": "\"If $\\\\Sigma a_n$ is divergent then it is not absolutely convergent.
\"", "templateType": "long string", "group": "Ungrouped variables", "name": "tr20", "description": ""}, "ch5": {"definition": "if(f=1,f1,if(f=2,f2,if(f=3,f3,f4)))", "templateType": "anything", "group": "Ungrouped variables", "name": "ch5", "description": ""}, "tr9": {"definition": "\"If $\\\\Sigma a_n$ is absolutely convergent then it is convergent.
\"", "templateType": "long string", "group": "Ungrouped variables", "name": "tr9", "description": ""}, "tr8": {"definition": "\"If $a_n \\\\neq 0$ for all $n \\\\in \\\\mathbb{N}$ and if $\\\\dfrac{a_{n+1}}{a_n}$ $\\\\to$ $\\\\ell$ with $|\\\\ell | <1$ as $n \\\\to \\\\infty$, then $\\\\Sigma a_n$ converges.
\"", "templateType": "long string", "group": "Ungrouped variables", "name": "tr8", "description": ""}, "f20": {"definition": "'It is not possible for an unbounded sequence to have a bounded subsequence.'", "templateType": "anything", "group": "Ungrouped variables", "name": "f20", "description": ""}, "tr1": {"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.
\"", "templateType": "long string", "group": "Ungrouped variables", "name": "tr1", "description": ""}, "ch4": {"definition": "if(w=1,tr13,if(w=2,tr14,if(w=3,tr15,tr16)))", "templateType": "anything", "group": "Ungrouped variables", "name": "ch4", "description": ""}, "tr3": {"definition": "\"If $a_n \\\\not\\\\to 0$ as $n \\\\to \\\\infty$, then the series $\\\\Sigma a_n$ diverges.
\"", "templateType": "long string", "group": "Ungrouped variables", "name": "tr3", "description": ""}, "tr2": {"definition": "\"If $a_n \\\\geq \\\\dfrac{1}{n}$ for all $n \\\\in \\\\mathbb{N}$, then $\\\\Sigma a_n$ diverges.
\"", "templateType": "long string", "group": "Ungrouped variables", "name": "tr2", "description": ""}, "tr5": {"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.
\"", "templateType": "long string", "group": "Ungrouped variables", "name": "tr5", "description": ""}, "tr4": {"definition": "\"If the series $\\\\Sigma a_n$ converges, then $a_n \\\\to 0$ as $n \\\\to \\\\infty$.
\"", "templateType": "long string", "group": "Ungrouped variables", "name": "tr4", "description": ""}, "tr7": {"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.
\"", "templateType": "long string", "group": "Ungrouped variables", "name": "tr7", "description": ""}, "tr6": {"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.
\"", "templateType": "long string", "group": "Ungrouped variables", "name": "tr6", "description": ""}, "tr15": {"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 $R \\\\geq |X|$.
\"", "templateType": "long string", "group": "Ungrouped variables", "name": "tr15", "description": ""}, "tr14": {"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 $|x|>R$.
\"", "templateType": "long string", "group": "Ungrouped variables", "name": "tr14", "description": ""}, "tr16": {"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 $R \\\\leq |X|$.
\"", "templateType": "long string", "group": "Ungrouped variables", "name": "tr16", "description": ""}, "tr11": {"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.
\"", "templateType": "long string", "group": "Ungrouped variables", "name": "tr11", "description": ""}, "tr10": {"definition": "\"If $\\\\Sigma a_n$ is divergent then it is not absolutely convergent.
\"", "templateType": "long string", "group": "Ungrouped variables", "name": "tr10", "description": ""}, "tr13": {"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 $|x|<R$.
\"", "templateType": "long string", "group": "Ungrouped variables", "name": "tr13", "description": ""}, "tr12": {"definition": "\"If $a_n = (-1)^{n-1} u_n$ with $u_n >0$ and if $\\{u_n\\}$ is increasing, then $\\\\Sigma a_n$ diverges.
\"", "templateType": "long string", "group": "Ungrouped variables", "name": "tr12", "description": ""}, "ch7": {"definition": "if(h=1,f9,if(h=2,f10,if(h=3,f11,f12)))", "templateType": "anything", "group": "Ungrouped variables", "name": "ch7", "description": ""}, "f12": {"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.
\"", "templateType": "long string", "group": "Ungrouped variables", "name": "f12", "description": ""}, "f13": {"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 $x=R$.
\"", "templateType": "long string", "group": "Ungrouped variables", "name": "f13", "description": ""}, "w": {"definition": "random(1..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "w", "description": ""}, "g": {"definition": "random(1..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "g", "description": ""}, "f": {"definition": "random(1..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "f", "description": ""}, "h": {"definition": "random(1..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "h", "description": ""}, "ch1": {"definition": "if(t=1,tr1,if(t=2,tr2,if(t=3,tr3,tr4)))", "templateType": "anything", "group": "Ungrouped variables", "name": "ch1", "description": ""}, "ch2": {"definition": "if(u=1,tr5,if(u=2,tr6,if(u=3,tr7,tr8)))", "templateType": "anything", "group": "Ungrouped variables", "name": "ch2", "description": ""}, "ch3": {"definition": "if(v=1,tr9,if(v=2,tr10,if(v=3,tr11,tr12)))", "templateType": "anything", "group": "Ungrouped variables", "name": "ch3", "description": ""}, "u": {"definition": "random(1..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "u", "description": ""}, "t": {"definition": "random(1..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "t", "description": ""}, "ch6": {"definition": "if(g=1,f5,if(g=6,f2,if(g=3,f7,f8)))", "templateType": "anything", "group": "Ungrouped variables", "name": "ch6", "description": ""}, "v": {"definition": "random(1..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "v", "description": ""}, "ch8": {"definition": "if(x=1,f13,if(x=2,f14,if(f=x,f15,f16)))", "templateType": "anything", "group": "Ungrouped variables", "name": "ch8", "description": ""}, "x": {"definition": "random(1..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "x", "description": ""}, "f10": {"definition": "\"If $\\\\Sigma a_n$ is not divergent then it is absolutely convergent.
\"", "templateType": "long string", "group": "Ungrouped variables", "name": "f10", "description": ""}, "f11": {"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.
\"", "templateType": "long string", "group": "Ungrouped variables", "name": "f11", "description": ""}, "f16": {"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 $n \\\\to \\\\infty$.
\"", "templateType": "long string", "group": "Ungrouped variables", "name": "f16", "description": ""}, "f14": {"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 $x=R$.
\"", "templateType": "long string", "group": "Ungrouped variables", "name": "f14", "description": ""}, "f15": {"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 $n \\\\to \\\\infty$.
\"", "templateType": "long string", "group": "Ungrouped variables", "name": "f15", "description": ""}}, "metadata": {"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.
\nCC By Newcastle University
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