[[0]]
\n \n \n \n", "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "gaps": [{"choices": ["{Ch1}
", "{Ch2}
", "{Ch3}
", "{Ch4}
", "{Ch5}
", "{Ch6}
", "{Ch7}
", "{Ch8}
"], "minAnswers": 0, "showFeedbackIcon": true, "shuffleChoices": true, "marks": 0, "shuffleAnswers": false, "maxMarks": 0, "displayType": "radiogroup", "variableReplacements": [], "showCorrectAnswer": true, "answers": [true, false], "scripts": {}, "maxAnswers": 0, "type": "m_n_x", "warningType": "none", "minMarks": 0, "variableReplacementStrategy": "originalfirst", "matrix": [[1, -1], [1, -1], [1, -1], ["1", "-1"], [-1, 1], [-1, 1], ["-1", "1"], ["-1", "1"]], "layout": {"type": "all", "expression": ""}}]}], "variable_groups": [], "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.
", "licence": "Creative Commons Attribution 4.0 International"}, "variables": {"f14": {"name": "f14", "group": "Ungrouped variables", "templateType": "long string", "description": "", "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 $x=R$.
\""}, "f2": {"name": "f2", "group": "Ungrouped variables", "templateType": "long string", "description": "", "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.
\""}, "h": {"name": "h", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "random(1..4)"}, "ch6": {"name": "ch6", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "if(g=1,f5,if(g=6,f2,if(g=3,f7,f8)))"}, "f7": {"name": "f7", "group": "Ungrouped variables", "templateType": "long string", "description": "", "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.
\""}, "f6": {"name": "f6", "group": "Ungrouped variables", "templateType": "long string", "description": "", "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.
\""}, "f": {"name": "f", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "random(1..4)"}, "f12": {"name": "f12", "group": "Ungrouped variables", "templateType": "long string", "description": "", "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.
\""}, "u": {"name": "u", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "random(1..4)"}, "f11": {"name": "f11", "group": "Ungrouped variables", "templateType": "long string", "description": "", "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.
\""}, "tr7": {"name": "tr7", "group": "Ungrouped variables", "templateType": "long string", "description": "", "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.
\""}, "f16": {"name": "f16", "group": "Ungrouped variables", "templateType": "long string", "description": "", "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 $n \\\\to \\\\infty$.
\""}, "tr6": {"name": "tr6", "group": "Ungrouped variables", "templateType": "long string", "description": "", "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.
\""}, "v": {"name": "v", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "random(1..4)"}, "tr8": {"name": "tr8", "group": "Ungrouped variables", "templateType": "long string", "description": "", "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.
\""}, "tr20": {"name": "tr20", "group": "Ungrouped variables", "templateType": "long string", "description": "", "definition": "\"If $\\\\Sigma a_n$ is divergent then it is not absolutely convergent.
\""}, "ch3": {"name": "ch3", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "if(v=1,tr9,if(v=2,tr10,if(v=3,tr11,tr12)))"}, "tr1": {"name": "tr1", "group": "Ungrouped variables", "templateType": "long string", "description": "", "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.
\""}, "f20": {"name": "f20", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "'It is not possible for an unbounded sequence to have a bounded subsequence.'"}, "ch8": {"name": "ch8", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "if(x=1,f13,if(x=2,f14,if(f=x,f15,f16)))"}, "f5": {"name": "f5", "group": "Ungrouped variables", "templateType": "long string", "description": "", "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.
\""}, "tr10": {"name": "tr10", "group": "Ungrouped variables", "templateType": "long string", "description": "", "definition": "\"If $\\\\Sigma a_n$ is divergent then it is not absolutely convergent.
\""}, "tr13": {"name": "tr13", "group": "Ungrouped variables", "templateType": "long string", "description": "", "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 $|x|<R$.
\""}, "ch4": {"name": "ch4", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "if(w=1,tr13,if(w=2,tr14,if(w=3,tr15,tr16)))"}, "ch2": {"name": "ch2", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "if(u=1,tr5,if(u=2,tr6,if(u=3,tr7,tr8)))"}, "ch1": {"name": "ch1", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "if(t=1,tr1,if(t=2,tr2,if(t=3,tr3,tr4)))"}, "f4": {"name": "f4", "group": "Ungrouped variables", "templateType": "long string", "description": "", "definition": "\"If the series $\\\\Sigma a_n$ diverges, then $a_n \\\\not\\\\to 0$ as $n \\\\to \\\\infty$.
\""}, "t": {"name": "t", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "random(1..4)"}, "f8": {"name": "f8", "group": "Ungrouped variables", "templateType": "long string", "description": "", "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.
\""}, "f15": {"name": "f15", "group": "Ungrouped variables", "templateType": "long string", "description": "", "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 $n \\\\to \\\\infty$.
\""}, "tr12": {"name": "tr12", "group": "Ungrouped variables", "templateType": "long string", "description": "", "definition": "\"If $a_n = (-1)^{n-1} u_n$ with $u_n >0$ and if $\\{u_n\\}$ is increasing, then $\\\\Sigma a_n$ diverges.
\""}, "tr14": {"name": "tr14", "group": "Ungrouped variables", "templateType": "long string", "description": "", "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 $|x|>R$.
\""}, "f9": {"name": "f9", "group": "Ungrouped variables", "templateType": "long string", "description": "", "definition": "\"If $\\\\Sigma a_n$ is convergent then it is absolutely convergent.
\""}, "tr9": {"name": "tr9", "group": "Ungrouped variables", "templateType": "long string", "description": "", "definition": "\"If $\\\\Sigma a_n$ is absolutely convergent then it is convergent.
\""}, "tr3": {"name": "tr3", "group": "Ungrouped variables", "templateType": "long string", "description": "", "definition": "\"If $a_n \\\\not\\\\to 0$ as $n \\\\to \\\\infty$, then the series $\\\\Sigma a_n$ diverges.
\""}, "x": {"name": "x", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "random(1..4)"}, "tr2": {"name": "tr2", "group": "Ungrouped variables", "templateType": "long string", "description": "", "definition": "\"If $a_n \\\\geq \\\\dfrac{1}{n}$ for all $n \\\\in \\\\mathbb{N}$, then $\\\\Sigma a_n$ diverges.
\""}, "f13": {"name": "f13", "group": "Ungrouped variables", "templateType": "long string", "description": "", "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 $x=R$.
\""}, "f10": {"name": "f10", "group": "Ungrouped variables", "templateType": "long string", "description": "", "definition": "\"If $\\\\Sigma a_n$ is not divergent then it is absolutely convergent.
\""}, "ch7": {"name": "ch7", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "if(h=1,f9,if(h=2,f10,if(h=3,f11,f12)))"}, "tr16": {"name": "tr16", "group": "Ungrouped variables", "templateType": "long string", "description": "", "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 $R \\\\leq |X|$.
\""}, "g": {"name": "g", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "random(1..4)"}, "tr5": {"name": "tr5", "group": "Ungrouped variables", "templateType": "long string", "description": "", "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.
\""}, "w": {"name": "w", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "random(1..4)"}, "f3": {"name": "f3", "group": "Ungrouped variables", "templateType": "long string", "description": "", "definition": "\"If $a_n \\\\to 0$ as $n \\\\to \\\\infty$, then the series $\\\\Sigma a_n$ converges.
\""}, "ch5": {"name": "ch5", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "if(f=1,f1,if(f=2,f2,if(f=3,f3,f4)))"}, "tr11": {"name": "tr11", "group": "Ungrouped variables", "templateType": "long string", "description": "", "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.
\""}, "tr4": {"name": "tr4", "group": "Ungrouped variables", "templateType": "long string", "description": "", "definition": "\"If the series $\\\\Sigma a_n$ converges, then $a_n \\\\to 0$ as $n \\\\to \\\\infty$.
\""}, "tr15": {"name": "tr15", "group": "Ungrouped variables", "templateType": "long string", "description": "", "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 $R \\\\geq |X|$.
\""}, "f1": {"name": "f1", "group": "Ungrouped variables", "templateType": "long string", "description": "", "definition": "\"If $a_n \\\\geq \\\\dfrac{1}{n^2}$ for all $n \\\\in \\\\mathbb{N}$, then $\\\\Sigma a_n$ converges.
\""}}, "extensions": [], "preamble": {"js": "", "css": ""}, "advice": "You should be able to work out the correct answers from your notes.
", "variablesTest": {"maxRuns": 100, "condition": ""}, "statement": "Answer the following question on series. Note that every correct answer is worth 1 mark, but every wrong answer loses a mark.
", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "tags": [], "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"], "functions": {}, "type": "question", "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}]}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}