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.
\"", "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}], "name": "True/false statements about convergent and divergent series, ", "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"displayType": "radiogroup", "layout": {"expression": ""}, "choices": ["{Ch1}
", "{Ch2}
", "{Ch3}
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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.
", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}