// Numbas version: exam_results_page_options
{"name": "Series Homework 2 (power series)", "metadata": {"description": "<p>Various question on power series&nbsp;</p>", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "duration": 0, "percentPass": 0, "showQuestionGroupNames": false, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", "", "", "", "", ""], "variable_overrides": [[], [], [], [], [], [], []], "questions": [{"name": "Ugur's copy of True/false statements about convergent and divergent series,", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18261/"}], "variable_groups": [], "variables": {"tr8": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"<p><span>If $a_n \\\\neq 0$</span><span>&nbsp; for all $n \\\\in \\\\mathbb{N}$ and if $\\\\dfrac{a_{n+1}}{a_n}$&nbsp;</span><span>$\\\\to$ $\\\\ell$ with $|\\\\ell | &lt;1$&nbsp;</span><span>as $n \\\\to \\\\infty$</span><span>, then $\\\\Sigma a_n$ converges.</span></p>\"", "name": "tr8", "description": ""}, "x": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "name": "x", "description": ""}, "f1": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"<p>If $a_n \\\\geq \\\\dfrac{1}{n^2}$ for all $n \\\\in \\\\mathbb{N}$, then $\\\\Sigma a_n$ converges.</p>\"", "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": "\"<p>If a power series $\\\\Sigma a_n x^n$ has radius of convergence $R$ with $0 &lt; R &lt;\\\\infty$, then the power series converges for all $x$ with $|x|&lt;R$.&nbsp;</p>\"", "name": "tr13", "description": ""}, "tr3": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"<p>If $a_n \\\\not\\\\to 0$ as $n \\\\to \\\\infty$, then the series $\\\\Sigma a_n$ diverges.</p>\"", "name": "tr3", "description": ""}, "f4": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"<p><span>If the series $\\\\Sigma a_n$ diverges, then $a_n \\\\not\\\\to 0$ as $n \\\\to \\\\infty$</span><span>.</span></p>\"", "name": "f4", "description": ""}, "f3": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"<p><span>If &nbsp;$a_n \\\\to 0$ as $n \\\\to \\\\infty$, then the series $\\\\Sigma a_n$ converges.</span></p>\"", "name": "f3", "description": ""}, "tr2": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"<p><span>If $a_n \\\\geq \\\\dfrac{1}{n}$ for all $n \\\\in \\\\mathbb{N}$, then $\\\\Sigma a_n$ diverges.</span></p>\"", "name": "tr2", "description": ""}, "f15": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"<p>If a power series $\\\\Sigma a_n x^n$ has radius of convergence $R$ with $0&lt;R&lt;\\\\infty$, then $a_n \\\\neq 0$ and $\\\\dfrac{a_{n+1}}{a_n}$ tends to a limit as $n \\\\to \\\\infty$.</p>\"", "name": "f15", "description": ""}, "f9": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"<p><span>If $\\\\Sigma a_n$ is convergent then it is absolutely convergent.</span></p>\"", "name": "f9", "description": ""}, "tr11": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"<p><span>If $a_n = (-1)^{n-1} u_n$ with $u_n &gt;0$ and if $\\{u_n\\}$ does not converge to $0$, then $\\\\Sigma a_n$ diverges.</span></p>\"", "name": "tr11", "description": ""}, "f10": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"<p><span>If $\\\\Sigma a_n$ is not divergent then it is absolutely convergent.</span></p>\"", "name": "f10", "description": ""}, "tr6": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"<p><span>If $a_n&gt;0$</span><span>&nbsp;for all $n \\\\in \\\\mathbb{N}$ and if $\\\\dfrac{a_{n+1}}{a_n} \\\\to \\\\ell &gt;1$&nbsp;</span><span>as $n \\\\to \\\\infty$</span><span>, then $\\\\Sigma a_n$ diverges.</span></p>\"", "name": "tr6", "description": ""}, "tr16": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"<p><span>If a power series $\\\\Sigma a_n x^n$ di</span><span>verges for some value $x=X \\\\neq 0$, then the radius of convergence $R$ satisfies</span><span>&nbsp;$R \\\\leq |X|$</span><span>.&nbsp;</span></p>\"", "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": "\"<p><span>If a power series $\\\\Sigma a_n x^n$ has radius of convergence $R$ with $0 &lt; R &lt;\\\\infty$, then the power series converges for $x=R$.&nbsp;</span></p>\"", "name": "f13", "description": ""}, "f14": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"<p><span>If a power series $\\\\Sigma a_n x^n$ has radius of convergence $R$ with $0 &lt; R &lt;\\\\infty$, then the power series diverges for $x=R$</span><span>.&nbsp;</span></p>\"", "name": "f14", "description": ""}, "tr1": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"<p><span>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.</span></p>\"", "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": "\"<p>If $a_n = (-1)^{n-1} u_n$ with $u_n &gt;0$ and if $\\{u_n\\}$ is not decreasing, then $\\\\Sigma a_n$ diverges.</p>\"", "name": "f11", "description": ""}, "v": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "name": "v", "description": ""}, "tr14": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"<p><span>If a power series $\\\\Sigma a_n x^n$ has radius of convergence $R$ with $0 &lt; R &lt;\\\\infty$, then the power series diverges for all $x$ with $|x|&gt;R$.&nbsp;</span></p>\"", "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": "\"<p><span>If $\\\\Sigma a_n$ is divergent then it is not absolutely convergent.</span></p>\"", "name": "tr10", "description": ""}, "t": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "name": "t", "description": ""}, "tr12": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"<p><span>If $a_n = (-1)^{n-1} u_n$ with $u_n &gt;0$ and if $\\{u_n\\}$ </span><span>is increasing</span><span>, then $\\\\Sigma a_n$ diverges.</span></p>\"", "name": "tr12", "description": ""}, "tr20": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"<p><span>If $\\\\Sigma a_n$ is divergent then it is not absolutely convergent.</span></p>\"", "name": "tr20", "description": ""}, "f7": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"<p><span>If $a_n &gt;&nbsp;0$ for all $n \\\\in \\\\mathbb{N}$ and if $\\\\dfrac{a_{n+1}}{a_n}$ $\\\\to$ &nbsp;$\\\\ell &gt;0$</span><span>&nbsp;as $n \\\\to \\\\infty$, then $\\\\Sigma a_n$ converges.</span></p>\"", "name": "f7", "description": ""}, "tr7": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"<p><span>If $a_n \\\\neq 0$</span><span>&nbsp;for all $n \\\\in \\\\mathbb{N}$ and if $\\\\dfrac{a_{n+1}}{a_n}$</span><span>&nbsp;$\\\\to$ $\\\\infty$ </span><span>as $n \\\\to \\\\infty$</span><span>, then $\\\\Sigma a_n$ diverges.</span></p>\"", "name": "tr7", "description": ""}, "tr9": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"<p>If $\\\\Sigma a_n$ is absolutely convergent then it is convergent.</p>\"", "name": "tr9", "description": ""}, "h": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "name": "h", "description": ""}, "f2": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"<p><span>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.</span></p>\"", "name": "f2", "description": ""}, "tr15": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"<p><span>If a power series $\\\\Sigma a_n x^n$ converges for some value $x=X \\\\neq 0$, then the radius of convergence $R$ satisfies&nbsp;$R \\\\geq |X|$.&nbsp;</span></p>\"", "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": "\"<p><span>If $a_n &gt;&nbsp;0$ for all $n \\\\in \\\\mathbb{N}$ and if $\\\\dfrac{a_{n+1}}{a_n} \\\\to \\\\ell &lt;1$ as $n \\\\to \\\\infty$, then $\\\\Sigma a_n$ converges.</span></p>\"", "name": "tr5", "description": ""}, "f5": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"<p><span>If $a_n &gt;&nbsp;0$ 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.</span></p>\"", "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": "\"<p><span>If $a_n \\\\neq 0$ for all $n \\\\in \\\\mathbb{N}$&nbsp;</span><span>and if $\\\\dfrac{a_{n+1}}{a_n}$ $\\\\to$ $\\\\ell&lt;1$</span><span>&nbsp;as $n \\\\to \\\\infty$, then $\\\\Sigma a_n$ converges.</span></p>\"", "name": "f8", "description": ""}, "f12": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"<p><span>If $a_n = (-1)^{n-1} u_n$ with $u_n \\\\geq&nbsp;0$ and if $\\{u_n\\}$ </span><span>is increasing</span><span>, then $\\\\Sigma a_n$ diverges.</span></p>\"", "name": "f12", "description": ""}, "f6": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"<p><span>If $a_n&gt;0$ for all</span><span>&nbsp;$n \\\\in \\\\mathbb{N}$ and if $\\\\dfrac{a_{n+1}}{a_n}$ $\\\\to$ $\\\\ell=1$&nbsp;</span><span>as $n \\\\to \\\\infty$</span><span>, then $\\\\Sigma a_n$ diverges.</span></p>\"", "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": "\"<p><span>If <span>the series </span><span>$\\\\Sigma a_n$ converges, then&nbsp;</span>$a_n \\\\to 0$ as $n \\\\to \\\\infty$</span><span>.</span></p>\"", "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": "\"<p><span>If a power series $\\\\Sigma a_n x^n$ with $a_n \\\\neq 0$ for all $n$ has radius of convergence $R$ with $R&lt;\\\\infty$, then $\\\\dfrac{a_{n+1}}{a_n}$ tends to a limit as $n \\\\to \\\\infty$.</span></p>\"", "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": ["<p>{Ch1}</p>", "<p>{Ch2}</p>", "<p>{Ch3}</p>", "<p>{Ch4}</p>", "<p>{Ch5}</p>", "<p>{Ch6}</p>", "<p>{Ch7}</p>", "<p>{Ch8}</p>"], "matrix": [[1, -1], [1, -1], [1, -1], ["1", "-1"], [-1, 1], [-1, 1], ["-1", "1"], ["-1", "1"]], "minAnswers": 0, "maxAnswers": 0, "shuffleChoices": true, "warningType": "none", "scripts": {}, "maxMarks": 0, "type": "m_n_x", "minMarks": 0, "shuffleAnswers": false, "showCorrectAnswer": true, "marks": 0, "answers": [true, false]}], "type": "gapfill", "prompt": "\n  \n  \n  <p>[[0]]</p>\n  \n  \n  \n", "showCorrectAnswer": true, "marks": 0}], "statement": "<p>Answer the following question on series. Note that every correct answer is worth 1 mark, but every wrong answer loses a mark.</p>", "tags": ["checked2015", "divergent series", "limits", "MAS1601", "MAS2224", "power series", "series"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>17/04/2015:</strong></p>\n<p>(OK)&nbsp;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).</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>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.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>You should be able to work out the correct answers from your notes.</p>"}, {"name": "radius of convergence -various", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18261/"}], "tags": [], "metadata": {"description": "<p>questions on radius of convergence</p>", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "<p>Find the radius of convergence of the following power series:</p>", "advice": "<p>Recall that for a given power series <br/>\\[\\displaystyle\\sum_{n=1}^\\infty a_n,\\]</p>\n<p>one can use <br/>\\[\\displaystyle\\lim_{n\\to\\infty}\\left| \\frac{a_n}{a_{n+1}}\\right|\\]&nbsp;</p>\n<p>to find its radius of convergence.</p>\n<p></p>\n<p>a) The radius of convergence is&nbsp;</p>\n<p>\\[\\displaystyle\\lim_{n\\to\\infty}\\left| \\frac{a_n}{a_{n+1}}\\right| = \\lim_{n\\to\\infty}\\left| \\frac{\\frac{(-1)^{n+1}}{n}}{\\frac{(-1)^{n+2}}{n+1}}\\right| <br/>= \\lim_{n\\to\\infty}\\frac{\\frac{1}{n}}{\\frac{1}{n+1}} = 1.\\]</p>\n<p></p>\n<p>So the power series is convergent when $|x|&lt;1$. Next we need to analyse the behaviour at the end points to determine the convergence.&nbsp;</p>\n<ul>\n<li>At $x=1$ the series becomes&nbsp; $\\displaystyle\\sum_{n=1}^\\infty \\frac{(-1)^{n+1}}{n}$, which, by alternating series test is convergent.&nbsp;</li>\n<li>At $x=-1$&nbsp;the series becomes&nbsp; $\\displaystyle\\sum_{n=1}^\\infty \\frac{(-1)^{2n+1}}{n}$, which is again convergent by alternating series test.</li>\n</ul>\n<p>Therefore the interval of convergence is $[-1,1]$ (or $-1\\leq x \\leq 1$).&nbsp;</p>\n<p></p>\n<p>b) The radius of convergence is&nbsp;</p>\n<p>\\[\\displaystyle\\lim_{n\\to\\infty}\\left| \\frac{a_n}{a_{n+1}}\\right| = \\lim_{n\\to\\infty}\\left| \\frac{\\frac{1}{(2n)!}}{\\frac{1}{(2n+2)!}}\\right| <br/>= \\lim_{n\\to\\infty} (2n+2)(2n+1) = \\infty.\\]</p>\n<p></p>\n<p>So the power series is convergent for all $x\\in\\mathbb R$.&nbsp;</p>\n<p></p>\n<p>c) The radius of convergence is&nbsp;</p>\n<p>\\[\\displaystyle\\lim_{n\\to\\infty}\\left| \\frac{a_n}{a_{n+1}}\\right| = \\lim_{n\\to\\infty}\\left| \\frac{(-1)^{n+1}}{(-1)^{n+2}}\\right| <br/>=&nbsp; 1.\\]</p>\n<p></p>\n<p>So the power series is convergent when $|x|&lt;1$. Next we need to analyse the behaviour at the end points to determine the convergence.&nbsp;</p>\n<ul>\n<li>At $x=1$ the series becomes&nbsp; $\\displaystyle\\sum_{n=1}^\\infty (-1)^{n+1}$, which is divergent.&nbsp;</li>\n<li>At $x=-1$ the series becomes&nbsp; $\\displaystyle\\sum_{n=1}^\\infty (-1)^{n+1}$, which is divergent.</li>\n</ul>\n<p>Therefore the interval of convergence is $(-1,1)$ (or $-1&lt; x &lt; 1$). </p>\n<p></p>\n<p></p>\n<p>d) The radius of convergence is&nbsp;</p>\n<p>\\[\\displaystyle\\lim_{n\\to\\infty}\\left| \\frac{a_n}{a_{n+1}}\\right| = \\lim_{n\\to\\infty}\\left| \\frac{\\frac{n}{5^{n-1}}}{\\frac{(n+1)}{5^{n}}}\\right| <br/>= 5.\\]</p>\n<p></p>\n<p>So the power series is convergent when $|x+2|&lt;5$; equivalently when $-7 &lt; x &lt; 3$. Next we need to analyse the behaviour at the end points to determine the convergence.&nbsp;</p>\n<ul>\n<li>At $x=3$ the series becomes&nbsp; $\\displaystyle\\sum_{n=1}^\\infty n$, which is divergent.&nbsp;</li>\n<li>At $x=-7$ the series becomes&nbsp; $\\displaystyle\\sum_{n=1}^\\infty (-1)^{n}n$, which is divergent.</li>\n</ul>\n<p>Therefore the interval of convergence is $(-7,3)$ (or $-7&lt; x &lt; 3$).</p>\n<p></p>\n<p>e) The radius of convergence is&nbsp;</p>\n<p>\\[\\displaystyle\\lim_{n\\to\\infty}\\left| \\frac{a_n}{a_{n+1}}\\right| = \\lim_{n\\to\\infty}\\left| \\frac{\\frac{2^n}{3^n}}{\\frac{2^{n+1}}{3^{n+1}}}\\right| <br/>= \\frac{3}{2}.\\]</p>\n<p></p>\n<p>So the power series is convergent when $|x|&lt;\\frac{3}{2}$. Next we need to analyse the behaviour at the end points to determine the convergence.&nbsp;</p>\n<ul>\n<li>At $x= 3/2 $ the series becomes&nbsp; $\\displaystyle\\sum_{n=1}^\\infty 1$, which is divergent.&nbsp;</li>\n<li>At $x=-3/2$ the series becomes&nbsp; $\\displaystyle\\sum_{n=1}^\\infty (-1)^{n}$, which is divergent.</li>\n</ul>\n<p>Therefore the interval of convergence is $(-3/2,3/2)$ (or $-3/2&lt; x &lt; 3/2$).&nbsp;</p>\n<p></p>\n<p>f) The radius of convergence is&nbsp;</p>\n<p>\\[\\displaystyle\\lim_{n\\to\\infty}\\left| \\frac{a_n}{a_{n+1}}\\right| = \\lim_{n\\to\\infty}\\left| \\frac{\\frac{(-1)^n}{n^2+1}}{\\frac{(-1)^{n+1}}{(n+1)^2+1}}\\right| <br/>= \\lim_{n\\to\\infty}\\frac{(n+1)^2+1}{n^2+1} = 1.\\]</p>\n<p></p>\n<p>So the power series is convergent when $|x + 2| &lt; 1$, i.e. $-3&lt; x&lt; -1$. Next we need to analyse the behaviour at the end points to determine the convergence.&nbsp;</p>\n<ul>\n<li>At $x= -1 $ the series becomes&nbsp; $\\displaystyle\\sum_{n=1}^\\infty \\frac{(-1)^n}{n^2+1}$, which is convergent, by alternating series test.&nbsp;</li>\n<li>At $x=-3$ the series becomes&nbsp; $\\displaystyle\\sum_{n=1}^\\infty \\frac{1}{n^2+1}$, which is&nbsp; convergent by comparison test (as $\\frac{1}{n^2+1} &lt;\\frac{1}{n^2}$, and $\\displaystyle\\sum_{n=1}^\\infty\\frac{1}{n^2}$ converges).</li>\n</ul>\n<p>Therefore the interval of convergence is $[-3,-1]$ (or $-3\\leq x \\leq -1$).</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>The power series $\\displaystyle\\sum_{n=1}^\\infty \\frac{(-1)^{n+1}x^n}{n}$ has radius of convergence [[0]]</p>\n<p>The interval of convergence is [[1]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "1", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "-1<x<=1", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>The power series $\\displaystyle\\sum_{n=0}^\\infty \\frac{x^n}{(2n)!}$ has radius of convergence [[0]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "infinity", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>The power series $\\displaystyle\\sum_{n=1}^\\infty (-1)^{n+1}x^{2n}$ has radius of convergence [[0]]</p>\n<p>The interval of convergence is [[1]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "1", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "-1<x<1", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>The power series $\\displaystyle\\sum_{n=1}^\\infty \\frac{n(x+2)^n}{5^{n-1}}$ has radius of convergence [[0]]&nbsp;</p>\n<p>The interval of convergence is [[1]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "5", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "-7<x<3", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>The power series $\\displaystyle\\sum_{n=1}^\\infty \\frac{(2x)^n}{3^{n}}$ has radius of convergence [[0]].</p>\n<p>The interval of convergence is [[1]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "3/2", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "-1<x<1", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>The power series $\\displaystyle\\sum_{n=1}^\\infty \\frac{(-1)^n (x+2)^n}{n^2 + 1}$ has radius of convergence [[0]].</p>\n<p>The interval of convergence is [[1]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "1", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "-3 <= x <= -1", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "power series sine", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18261/"}], "tags": [], "metadata": {"description": "<p>power series expansions for sine</p>", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "<p>Find the power series expansion of $\\sin x$ by cnsidering the following.</p>", "advice": "<p>a) We know that $\\sin x$ has infitely many zeros. On the other hand a polynomial of degree $n$ can have at most $n$ roots. Hence $\\sin x$ cannot be a polynomial. (If we try to express $sin x $ as a polynomial, we see that the degree of the polynomial must be infinite. In other words, we need a power series!)</p>\n<p></p>\n<p>b) Suppose we have a polynomial&nbsp;</p>\n<p>\\[y(x) = a_0 + a_1x + a_2x^2 + a_3x^3 + \\ldots + a_nx^n\\]&nbsp;</p>\n<p>which is a solution to the given differential equation.&nbsp;</p>\n<p>Then, since $y^{\\prime\\prime} = - y$ we have&nbsp;</p>\n<p>\\[2a_2 + 6a_3x +12a_4x^2+\\ldots + n(n-1)a_nx^{n-2} =&nbsp; -a_0 - a_1x - a_2x^2 - a_3x^3 - \\ldots - a_nx^n .\\]</p>\n<p>Then</p>\n<ul>\n<li>$2a_2= -a_0=0$ (since y(0)=0), so $a_2=0$</li>\n<li>$6a_3 = -a_1 = -1$ (since y^prime(0)=1), so $a_3 = \\frac{-1}{3!}$.</li>\n<li>$12a_4 = -a_2 = 0$, so $a_4=0$.</li>\n<li>$20a_5 = -a_3 = \\frac{1}{5!}$.&nbsp;</li>\n<li>In general $a_n=0$ if $n$ is odd, and if $n$ is odd, $a_n=\\frac{-1}{n!}$ if $n$ or $a_n=\\frac{1}{n!}$ alternatingly&nbsp;</li>\n</ul>\n<p>But observe that the degrees of $y^{\\prime\\prime}$&nbsp; and $-y$ does not match. So, infact $y$ should be a power series:</p>\n<p>\\[y(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \\ldots + a_nx^n+\\ldots\\]</p>\n<p>with the coefficents computed as above. Discarding the zero terms we can rewrite it as&nbsp;</p>\n<p>\\[y(x) = -1+ -\\frac{1}{3!} x^3+ \\frac{1}{5!} x^5 -\\frac{1}{7!}x^7+\\frac{1}{9!}x^9 -\\frac{1}{11!}x^{11} + \\ldots.\\]</p>\n<p></p>\n<p>c) We know that $sin x$ is the only solution to the differential equation in part b) (why it is the only one?). So we must have&nbsp;</p>\n<p>\\[\\sin x =-1+ -\\frac{1}{3!} x^3+ \\frac{1}{5!} x^5 -\\frac{1}{7!}x^7+\\frac{1}{9!}x^9 -\\frac{1}{11!}x^{11} + + \\ldots.\\]</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "information", "useCustomName": true, "customName": "a)", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Explain why $\\sin x$ is not a polynomial (Hint: Think about how many zeros $sin x$ has).&nbsp;</p>"}, {"type": "information", "useCustomName": true, "customName": "b)", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Give an \"infinite polynomial\" solution to&nbsp;</p>\n<p>\\[\\frac{d^2 y}{dx^2} = -y; \\mbox{ and } y(0)=0, y^\\prime(0) = 1.\\]</p>\n<pre></pre>"}, {"type": "information", "useCustomName": true, "customName": "c)", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Hence express $\\sin x$ as an \"infinite polynomial\".</p>"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Ugur's copy of Taylor series (three terms)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18261/"}], "tags": ["3 term Taylor series", "approximation", "approximations", "Calculus", "calculus", "checked2015", "function", "functions", "series approximation", "series expansion", "taylor series", "Taylor series", "three term Taylor series"], "metadata": {"description": "<p>Find the first 3 terms in the Taylor series at $x=c$ for $f(x)=(a+bx)^{1/n}$ i.e. up to and including terms in $x^2$.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>find the first 3 terms in the Taylor series around $x=\\var{c}$ for $f(x)=(\\simplify[all]{{a-b*c}+{b}*x})^{1/\\var{n}}$ (i.e. up to terms in $x^2$).</p>", "advice": "<p>The first three terms in the Taylor series are given by $\\simplify[all]{a+b(x-{c})+c(x-{c})^2}$ where $\\displaystyle a=f(\\var{c}),\\;\\;b=f'(\\var{c}),\\;\\;c=\\frac{f''(\\var{c})}{2}$<br/>For this example, <br/>\\[\\begin{eqnarray*} f'(x)&amp;=&amp;\\simplify[all,fractionNumbers]{{b}/{n}*({a-b*c}+{b}x)^(-{n-1}/{n})}\\\\ f''(x)&amp;=&amp;\\simplify[all,fractionNumbers]{-{b^2*(n-1)}/{n^2}*({a-b*c}+{b}x)^(-{2*n-1}/{n})} \\end{eqnarray*} \\]<br/>and so we get:<br/>\\[\\begin{eqnarray*} a&amp;=&amp;f(\\var{c})=\\simplify[all]{{a}^(1/{n})={tm0}}\\\\ b&amp;=&amp;f'(\\var{c})=\\simplify[all,fractionNumbers]{{tm1}/{a*n}}\\\\ c&amp;=&amp;\\frac{f''(\\var{c})}{2}=\\simplify[all,fractionNumbers]{{tm2}/{2*a^2*n^2}} \\end{eqnarray*}\\]<br/>Hence the first three terms of the Taylor series are:<br/>\\[\\simplify[all,fractionNumbers,!collectNumbers]{{tm0}+{tm1}/{a*n}*(x-{c})+{tm2}/{2*a^2*n^2}*(x-{c})^2} \\]</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"s1": {"name": "s1", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything", "can_override": false}, "tm1": {"name": "tm1", "group": "Ungrouped variables", "definition": "tm0*b", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "if(a=4 or a=9 or a=25 or a=36 or a=49,2,if(a=8 or a=27,3,if(a=32,5,if(a=16,random(2,4),random(2..5)))))", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1,4,8,9,16,27,32,25,36,49)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "tm0": {"name": "tm0", "group": "Ungrouped variables", "definition": "a^(1/n)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "s1*switch(a=1,random(2..9),a=4,random(3,5,7,9),a=8,random(1,3,5,7,9),a=9,random(1,2,4,5,7,8),a=16,random(1,3,5,7,9),a=32,random(1,3,5,7,9),a=25,random(1,2,4,6,7,9),a=27,random(1,2,4,5,7,8),a=36,random(1,5,7,9),random(1,2,3,4,5,8,9))", "description": "", "templateType": "anything", "can_override": false}, "tm2": {"name": "tm2", "group": "Ungrouped variables", "definition": "-(n-1)*tm1*b", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "tm0", "tm2", "b", "tm1", "s1", "c", "n"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 4, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "\n        <p>Input the first three terms in the Taylor series in the form $a+b(x-\\var{c})+c(x-\\var{c})^2$ for suitable coefficients $a,\\;b$ and $c$.</p>\n        <p><br/>Input coefficients as fractions, not as decimals. Also do not use factorials in your answer. For example, input 6 rather than 3!.</p>\n        ", "answer": "{tm0}+{tm1}/{a*n}*(x-{c})+{tm2}/{2*a^2*n^2}*(x-{c})^2", "answerSimplification": "all,fractionNumbers,!collectNumbers", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 1e-06, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "notallowed": {"strings": ["!", "."], "showStrings": false, "partialCredit": 0, "message": "<p>Do not input factorials or decimals in the Taylor series.</p>"}, "valuegenerators": [{"name": "x", "value": ""}]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Ugur's copy of Maclaurin series (three terms)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18261/"}], "tags": ["approximate", "approximations", "Calculus", "calculus", "checked2015", "first three terms in a maclaurin series", "First three terms in MacLaurin series", "functions", "maclaurin series"], "metadata": {"description": "<p>Find the first 3 terms in the MacLaurin series for $f(x)=(a+bx)^{1/n}$ i.e. up to and including terms in $x^2$.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Find the first 3 terms in the MacLaurin series for $f(x)=(\\simplify[all]{{a}+{b}*x})^{1/\\var{n}}$ i.e. up to terms in $x^2$.</p>", "advice": "<p>The first three terms in the MacLaurin series are given by $a+bx+cx^2$ where $\\displaystyle a=f(0),\\;\\;b=f'(0),\\;\\;c=\\frac{f''(0)}{2}$<br/>For this example, <br/>\\[\\begin{eqnarray*} f'(x)&amp;=&amp;\\simplify[all,fractionNumbers]{{b}/{n}*({a}+{b}x)^(-{n-1}/{n})}\\\\ f''(x)&amp;=&amp;\\simplify[all,fractionNumbers]{-{b^2*(n-1)}/{n^2}*({a}+{b}x)^(-{2*n-1}/{n})} \\end{eqnarray*} \\]<br/>and so we get:<br/>\\[\\begin{eqnarray*} a&amp;=&amp;f(0)=\\simplify[all]{{a}^(1/{n})={tm0}}\\\\ b&amp;=&amp;f'(0)=\\simplify[all,fractionNumbers]{{tm1}/{a*n}}\\\\ c&amp;=&amp;\\frac{f''(0)}{2}=\\simplify[all,fractionNumbers]{{tm2}/{2*a^2*n^2}} \\end{eqnarray*}\\]<br/>Hence the first three terms of the MacLaurin series are:<br/>\\[\\simplify[all,fractionNumbers,!collectNumbers]{{tm0}+{tm1}/{a*n}*x+{tm2}/{2*a^2*n^2}*x^2} \\]</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"s1": {"name": "s1", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything", "can_override": false}, "tm1": {"name": "tm1", "group": "Ungrouped variables", "definition": "tm0*b", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "if(a=4 or a=9 or a=25 or a=36 or a=49,2,if(a=8 or a=27,3,if(a=32,5,if(a=16,random(2,4),random(2..5)))))", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "s1*switch(a=1,random(2..9),a=4,random(3,5,7,9),a=8,random(1,3,5,7,9),a=9,random(1,2,4,5,7,8),a=16,random(1,3,5,7,9),a=32,random(1,3,5,7,9),a=25,random(1,2,4,6,7,9),a=27,random(1,2,4,5,7,8),a=36,random(1,5,7,9),random(1,2,3,4,5,8,9))", "description": "", "templateType": "anything", "can_override": false}, "tm0": {"name": "tm0", "group": "Ungrouped variables", "definition": "a^(1/n)", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1,4,8,9,16,27,32,25,36,49)", "description": "", "templateType": "anything", "can_override": false}, "tm2": {"name": "tm2", "group": "Ungrouped variables", "definition": "-(n-1)*tm1*b", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "tm0", "tm2", "b", "tm1", "s1", "n"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 4, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>First 3 terms = ?<br/>Input coefficients as fractions, not as decimals. Also do not use factorials in your answer. For example, input 6 rather than 3!.</p>", "answer": "{tm0}+{tm1}/{a*n}*x+{tm2}/{2*a^2*n^2}*x^2", "answerSimplification": "all,fractionNumbers,!collectNumbers", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 1e-06, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Fourth order Taylor approximation of cos(2x)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18261/"}], "tags": [], "metadata": {"description": "<p>Taylor Approximation ofr $\\cos(2x)$</p>", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "<p>Fin the fourth order Taylor Approximation of $\\cos(2x)$</p>", "advice": "<p><strong>a)</strong></p>\n<p>We need the derivatives of $\\cos(2x)$ up to fourth order. They are&nbsp;</p>\n<p>\\begin{align}<br/>\\cos^{(0)}(2x) &amp;= \\cos(2x) \\quad&nbsp; &nbsp;&amp;\\cos^{(1)}(2x) &amp;= -2\\sin(2x)\\\\<br/>\\cos^{(2)}(2x) &amp;= -4\\cos(2x) \\quad&nbsp; &amp;\\cos^{(3)}(2x) &amp;= 8\\sin(2x)\\\\<br/>\\cos^{(4)}(2x) &amp;= 16\\cos(2x) \\quad&nbsp;&nbsp;<br/>\\end{align}</p>\n<p>Then the fourth order Taylor Approximation to the given function is&nbsp;</p>\n<p>\\[\\cos\\left(\\simplify[fractionNumbers, zeroFactor, unitFactor]{{r2}*pi}\\right) - <br/>2\\sin\\left(\\simplify[fractionNumbers, zeroFactor, unitFactor]{{r2}*pi}\\right)\\left(\\simplify[zeroTerm, fractionNumbers]{x-{r}*pi}\\right) - <br/>\\frac{4\\cos\\left(\\simplify[fractionNumbers, zeroFactor, unitFactor]{{r2}*pi}\\right)}{2}\\left(\\simplify[zeroTerm, fractionNumbers]{x-{r}*pi}\\right)^2 <br/>+\\frac{8\\sin\\left(\\simplify[fractionNumbers, zeroFactor, unitFactor]{{r2}*pi}\\right)}{3!}\\left(\\simplify[zeroTerm, fractionNumbers]{x-{r}*pi}\\right)^3 + <br/>\\frac{16\\cos\\left(\\simplify[fractionNumbers, zeroFactor, unitFactor]{{r2}*pi}\\right)}{4!}\\left(\\simplify[zeroTerm, fractionNumbers]{x-{r}*pi}\\right)^4 = <br/>\\\\[3mm]<br/>\\cos\\left(\\simplify[fractionNumbers, zeroFactor, unitFactor]{{r2}*pi}\\right) - <br/>2\\sin\\left(\\simplify[fractionNumbers, zeroFactor, unitFactor]{{r2}*pi}\\right)\\left(\\simplify[zeroTerm, fractionNumbers]{x-{r}*pi}\\right) - <br/>2\\cos\\left(\\simplify[fractionNumbers, zeroFactor, unitFactor]{{r2}*pi}\\right)\\left(\\simplify[zeroTerm, fractionNumbers]{x-{r}*pi}\\right)^2 + <br/>\\frac{4\\sin\\left(\\simplify[fractionNumbers, zeroFactor, unitFactor]{{r2}*pi}\\right)}{3}\\left(\\simplify[zeroTerm, fractionNumbers]{x-{r}*pi}\\right)^3 + <br/>\\frac{2\\cos\\left(\\simplify[fractionNumbers, zeroFactor, unitFactor]{{r2}*pi}\\right)}{3}\\left(\\simplify[zeroTerm, fractionNumbers]{x-{r}*pi}\\right)^4<br/>\\]</p>\n<p><strong>(Note that in the answer above, the algorithm approximates sine and cosine values that appear in the polynomial we gave here)</strong></p>\n<p><strong></strong></p>\n<p><strong>b)</strong></p>\n<p>We need the fifth derivative for the error term. That is $\\cos^{(5)}(2x) = -32\\sin(2x)$. Then by Taylor's Theorem the error can computed by&nbsp;</p>\n<p>\\[R_5(\\simplify[fractionNumbers, unitFactor, zeroFactor]{{r2}*pi}) = \\frac{(\\simplify[fractionNumbers, zeroFactor, unitFactor]{{r2}*pi - {r}*pi})}{5!}\\cos^{(5)}(2\\eta) = <br/>\\frac{(\\simplify[fractionNumbers, zeroFactor, unitFactor]{{r2}*pi - {r}*pi})}{5!}(-32(\\sin(2\\eta)) \\quad \\\\[1mm] \\mbox{for some } \\eta\\in \\left[\\simplify[fractionNumbers, unitFactor, zeroFactor]{{r}*pi}, \\simplify[fractionNumbers, unitFactor, zeroFactor]{{r2}*pi}\\right] \\quad (\\mbox{or }\\; <br/>\\eta\\in \\left[\\simplify[fractionNumbers, unitFactor, zeroFactor]{{r2}*pi}, \\simplify[fractionNumbers, unitFactor, zeroFactor]{{r}*pi}\\right])\\]</p>\n<p></p>\n<p>Note that findin what $\\eta$ should is not possible at this stage. However, one can still bound the error. For that you need to find the $\\eta \\in \\left[\\simplify[fractionNumbers, unitFactor, zeroFactor]{{r}*pi}, \\simplify[fractionNumbers, unitFactor, zeroFactor]{{r2}*pi}\\right]$ such that $\\sin(2\\eta)$ is maximal.</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"r": {"name": "r", "group": "Ungrouped variables", "definition": "random(0, -1/4, 1/3, 1/6, 1/4, 1/2)", "description": "", "templateType": "anything", "can_override": false}, "a0": {"name": "a0", "group": "Ungrouped variables", "definition": "cos(2*(r*pi))", "description": "", "templateType": "anything", "can_override": false}, "a1": {"name": "a1", "group": "Ungrouped variables", "definition": "-2*sin(2*(r*pi))", "description": "", "templateType": "anything", "can_override": false}, "a2": {"name": "a2", "group": "Ungrouped variables", "definition": "-2*cos(2*(r*pi))", "description": "", "templateType": "anything", "can_override": false}, "a3": {"name": "a3", "group": "Ungrouped variables", "definition": "4*sin(2*(r*pi))/3", "description": "", "templateType": "anything", "can_override": false}, "a4": {"name": "a4", "group": "Ungrouped variables", "definition": "2*cos(2*(r*pi))/3", "description": "", "templateType": "anything", "can_override": false}, "r1": {"name": "r1", "group": "Ungrouped variables", "definition": "random([0, -1/4, 1/3, 1/6, 1/4, 1/2] except {r}) ", "description": "", "templateType": "anything", "can_override": false}, "r2": {"name": "r2", "group": "Ungrouped variables", "definition": "2*r", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["r", "a0", "a1", "a2", "a3", "a4", "r1", "r2"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Find the $4$-th order Taylor Approximation to $\\cos x$ around $x=\\simplify[zeroFactor, fractionNumbers]{{r}*pi}$.</p>\n<p>[[0]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{a0} + {a1}*(x-{r}*pi) + {a2}*(x-{r}*pi)^2 + {a3}*(x-{r}*pi)^3 + {a4}*(x-{r}*pi)^4", "answerSimplification": "basic, trig, zeroFactor, zeroTerm, collectNumbers, simplifyFractions", "showPreview": true, "checkingType": "dp", "checkingAccuracy": 3, "failureRate": 1, "vsetRangePoints": "2", "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}, {"type": "information", "useCustomName": true, "customName": "b)", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Find an upper bound for the error between the approximation you found above and $\\cos(2x)$ at $x=\\simplify[zeroFactor, unitFactor, fractionNumbers]{{r2}*pi}$.</p>"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Power series solution of second order ODE", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..3)*sign(random(-1,1))", "name": "b1", "description": ""}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..3)*sign(random(-1,1))", "name": "a1", "description": ""}, "amp2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "'$a_{m+2}$'", "name": "amp2", "description": ""}}, "ungrouped_variables": ["a1", "amp2", "b1"], "rulesets": {}, "functions": {"addsumfunction": {"type": "string", "language": "javascript", "definition": "Numbas.jme.display.texOps.sum=function(thing,texArgs) {\n        var lowerLimit = '';\n        var upperLimit = '';\n        if (texArgs[1]) {\n      lowerLimit = texArgs[1];\n        }\n        if (texArgs[2]) {\n      upperLimit = texArgs[2];\n        }\n        return ('\\\\sum_{'+lowerLimit+'}^{'+upperLimit+'}{'+texArgs[0]+'}');\n      }\n      return ('');", "parameters": []}}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"answer": "{-b1}*a0/2", "showCorrectAnswer": true, "vsetrange": [0, 1], "scripts": {}, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"message": "<p>Do not enter decimals in your answer.</p>", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "checkingaccuracy": 0.001, "type": "jme", "answersimplification": "all", "marks": 1, "vsetrangepoints": 5}, {"answer": "-{a1+b1}*a1/6", "showCorrectAnswer": true, "vsetrange": [0, 1], "scripts": {}, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"message": "<p>Do not enter decimals in your answer.</p>", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "checkingaccuracy": 0.001, "type": "jme", "answersimplification": "all", "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "<p>In your answer use the symbols&nbsp;<code>a0</code> and <code>a1</code> for $a_0$ and&nbsp;$a_1$ respectively. &nbsp;In addition, do not enter decimals.</p>\n        <p>$a_2=$ [[0]].</p>\n        <p>$a_3=$ [[1]].</p>", "marks": 0}], "statement": "<p>{addSumFunction()}</p>\n  <p>Seek a power series solution, about $x=0$, in the form</p>\n  <p>\\[y(x)=\\sum_{n=0}^{\\infty}{a_nx^n},\\]</p>\n  <p>of the differential equation</p>\n  <p>\\[\\simplify{y''+{a1}*x*y'+{b1}*y}=0.\\]</p>\n  <p>Take $a_0$ and $a_1$ to be arbitrary constants, and enter the coefficients $a_2$ and $a_3$ as functions of $a_0$ and $a_1$.</p>", "tags": ["checked2015", "MAS2105"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p>Uses a custom function to allow simplification of a LaTeX sum, in the same manner as e.g. <code>int()</code> or <code>defint()</code>.</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Power series solution of $y''+axy'+by=0$ about $x=0$.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>We have</p>\n<p>\\[y(x)=\\sum_{n=0}^{\\infty}{a_nx^n},\\]</p>\n<p>so</p>\n<p>\\[y'(x)=\\sum_{n=1}^{\\infty}{a_nnx^{n-1}},\\]</p>\n<p>and</p>\n<p>\\[y''(x)=\\sum_{n=2}^{\\infty}{a_nn(n-1)x^{n-2}}.\\]</p>\n<p>Substitute these expressions into the original differential equation to obtain</p>\n<p>\\[\\simplify{sum(a_n*n*(n-1)x^(n-2),n=2,infty)+{a1}*sum(a_n*n*x^n,n=1,infty)+{b1}*sum(a_n*x^n,n=0,infty)}=0.\\]</p>\n<p>Now reset the index $m=n-2$ in the first summation, and $m=n$ in the second and third summations to obtain</p>\n<p>\\[\\simplify{sum({amp2}*(m+2)*(m+1)x^m,m=0,infty)+{a1}*sum(a_m*m*x^m,m=1,infty)+{b1}*sum(a_m*x^m,m=0,infty)}=0.\\]</p>\n<p>This equation must be valid for all values of $x$, so the coefficients of like powers of $x$ must vanish. &nbsp;Take $m=0$ to obtain the coefficients of $x^0$, then</p>\n<p>\\[\\simplify{2*a2+{b1}*a0}=0,\\]</p>\n<p>and so</p>\n<p>\\[a_2=\\simplify{{-b1}*a0/2}.\\]</p>\n<p>Now take $m=1$ to obtain the coefficients of $x^1$, so</p>\n<p>\\[\\simplify{6*a3+{a1}*a1+{b1}*a1}=0,\\]</p>\n<p>then</p>\n<p>\\[a_3=\\simplify{-{a1+b1}*a1/6}.\\]</p>"}]}], "allowPrinting": true, "navigation": {"allowregen": true, "reverse": true, "browse": true, "allowsteps": true, "showfrontpage": true, "showresultspage": "oncompletion", "navigatemode": "sequence", "onleave": {"action": "none", "message": ""}, "preventleave": true, "startpassword": "", "allowAttemptDownload": false, "downloadEncryptionKey": ""}, "timing": {"allowPause": true, "timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "feedback": {"showactualmark": true, "showtotalmark": true, "showanswerstate": true, "allowrevealanswer": true, "advicethreshold": 0, "intro": "", "end_message": "", "reviewshowscore": true, "reviewshowfeedback": true, "reviewshowexpectedanswer": true, "reviewshowadvice": true, "feedbackmessages": []}, "diagnostic": {"knowledge_graph": {"topics": [], "learning_objectives": []}, "script": "diagnosys", "customScript": ""}, "contributors": [{"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18261/"}], "extensions": [], "custom_part_types": [], "resources": []}