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{"name": "Series: p series", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"statement": "<p>You are given the series</p>\n<p>\\[\\sum_{k=\\var{start}}^\\infty \\simplify[fractionNumbers]{{num}/({den}*k^{p})}.\\]</p>", "preamble": {"css": "", "js": ""}, "variables": {"den": {"definition": "random([2,3,5,7,11,13] except [num,-num])", "group": "Ungrouped variables", "description": "", "templateType": "anything", "name": "den"}, "p": {"definition": "random(1,2,3,4,5,6,7,8,1/2,1/3,1/4,1/5,1/6,1/7,1/8)", "group": "Ungrouped variables", "description": "", "templateType": "anything", "name": "p"}, "start": {"definition": "random(list(2..7) except [num,den])", "group": "Ungrouped variables", "description": "", "templateType": "anything", "name": "start"}, "num": {"definition": "random(2,3,5,7,11,13,-2,-3,-5,-7,-11,-13)", "group": "Ungrouped variables", "description": "", "templateType": "anything", "name": "num"}}, "rulesets": {}, "advice": "<p>The $p$-series $\\sum_{n=1}^\\infty \\frac{1}{n^p}$ is convergent if $p&gt;1$ and divergent if $p\\le 1$.</p>\n<p>Our series is $\\sum_{k=\\var{start}}^\\infty \\simplify[fractionNumbers]{{num}/({den}*k^{p})}$, which looks similar with a $p$-value of $\\var[fractionNumbers]{p}$.&nbsp;In fact:</p>\n<p><span data-jme-visible=\"start=2\">\\[\\begin{align}\\sum_{k=\\var{start}}^\\infty \\simplify[fractionNumbers]{{num}/({den}*k^{p})}&amp;=\\simplify[fractionNumbers]{{num}/{den}}\\sum_{k=\\var{start}}^\\infty \\simplify[fractionNumbers]{1/(k^{p})}\\\\&amp;=\\simplify[fractionNumbers]{{num}/{den}}\\left(-1+\\sum_{k=1}^\\infty \\simplify[fractionNumbers]{1/k^{p}}\\right)\\end{align}\\]</span></p>\n<p><span data-jme-visible=\"start=3\">\\[\\begin{align}\\sum_{k=\\var{start}}^\\infty \\simplify[fractionNumbers]{{num}/({den}*k^{p})}&amp;=\\simplify[fractionNumbers]{{num}/{den}}\\sum_{k=\\var{start}}^\\infty \\simplify[fractionNumbers]{1/(k^{p})}\\\\&amp;=\\simplify[fractionNumbers]{{num}/{den}}\\left(-1-\\frac{1}{2^\\var[fractionNumbers]{p}}+\\sum_{k=1}^\\infty \\simplify[fractionNumbers]{1/k^{p}}\\right)\\end{align}\\]&nbsp;</span></p>\n<p><span data-jme-visible=\"start=4\">\\[\\begin{align}\\sum_{k=\\var{start}}^\\infty \\simplify[fractionNumbers]{{num}/({den}*k^{p})}&amp;=\\simplify[fractionNumbers]{{num}/{den}}\\sum_{k=\\var{start}}^\\infty \\simplify[fractionNumbers]{1/(k^{p})}\\\\&amp;=\\simplify[fractionNumbers]{{num}/{den}}\\left(-1-\\frac{1}{2^\\var[fractionNumbers]{p}}-\\frac{1}{3^\\var[fractionNumbers]{p}}+\\sum_{k=1}^\\infty \\simplify[fractionNumbers]{1/k^{p}}\\right)\\end{align}\\]</span>&nbsp;</p>\n<p><span data-jme-visible=\"start=5\">\\[\\begin{align}\\sum_{k=\\var{start}}^\\infty \\simplify[fractionNumbers]{{num}/({den}*k^{p})}&amp;=\\simplify[fractionNumbers]{{num}/{den}}\\sum_{k=\\var{start}}^\\infty \\simplify[fractionNumbers]{1/(k^{p})}\\\\&amp;=\\simplify[fractionNumbers]{{num}/{den}}\\left(-1-\\frac{1}{2^\\var[fractionNumbers]{p}}-\\frac{1}{3^\\var[fractionNumbers]{p}}-\\frac{1}{4^\\var[fractionNumbers]{p}}+\\sum_{k=1}^\\infty \\simplify[fractionNumbers]{1/k^{p}}\\right)\\end{align}\\]&nbsp;</span></p>\n<p><span data-jme-visible=\"start=6\">\\[\\begin{align}\\sum_{k=\\var{start}}^\\infty \\simplify[fractionNumbers]{{num}/({den}*k^{p})}&amp;=\\simplify[fractionNumbers]{{num}/{den}}\\sum_{k=\\var{start}}^\\infty \\simplify[fractionNumbers]{1/(k^{p})}\\\\&amp;=\\simplify[fractionNumbers]{{num}/{den}}\\left(-1-\\frac{1}{2^\\var[fractionNumbers]{p}}-\\frac{1}{3^\\var[fractionNumbers]{p}}-\\frac{1}{4^\\var[fractionNumbers]{p}}-\\frac{1}{5^\\var[fractionNumbers]{p}}+\\sum_{k=1}^\\infty \\simplify[fractionNumbers]{1/k^{p}}\\right)\\end{align}\\]&nbsp;</span></p>\n<p><span data-jme-visible=\"start=7\">\\[\\begin{align}\\sum_{k=\\var{start}}^\\infty \\simplify[fractionNumbers]{{num}/({den}*k^{p})}&amp;=\\simplify[fractionNumbers]{{num}/{den}}\\sum_{k=\\var{start}}^\\infty \\simplify[fractionNumbers]{1/(k^{p})}\\\\&amp;=\\simplify[fractionNumbers]{{num}/{den}}\\left(-1-\\frac{1}{2^\\var[fractionNumbers]{p}}-\\frac{1}{3^\\var[fractionNumbers]{p}}-\\frac{1}{4^\\var[fractionNumbers]{p}}-\\frac{1}{5^\\var[fractionNumbers]{p}}-\\frac{1}{6^\\var[fractionNumbers]{p}}+\\sum_{k=1}^\\infty \\simplify[fractionNumbers]{1/k^{p}}\\right)\\end{align}\\]&nbsp;</span></p>\n<p>and so the convergence/divergence of our series depends on the convergence/divergence of the related $p$-series. Note that adding or subtracting a finite number of terms to a series will not change whether it converges or diverges (such as our missing first <span data-jme-visible=\"start=2\">term</span> <span data-jme-visible=\"start&lt;&gt;2\">terms</span>), nor will multiplying or dividing&nbsp;the series by a non-zero constant (such as $\\frac{\\var{num}}{\\var{den}}$).</p>\n<p>So we have a&nbsp;<span data-jme-visible=\"p&gt;1\"><g class=\"gr_ gr_5 gr-alert gr_spell gr_inline_cards gr_run_anim ContextualSpelling ins-del multiReplace\" id=\"5\" data-gr-id=\"5\">convergent</g></span><g class=\"gr_ gr_5 gr-alert gr_spell gr_inline_cards gr_disable_anim_appear ContextualSpelling ins-del multiReplace\" id=\"5\" data-gr-id=\"5\"> </g><span data-jme-visible=\"p&lt;=1\"><g class=\"gr_ gr_5 gr-alert gr_spell gr_inline_cards gr_disable_anim_appear ContextualSpelling ins-del multiReplace\" id=\"5\" data-gr-id=\"5\">divergent</g></span> series that you could say is really a $p$-series with $p=\\var[fractionNumbers]{p}$.</p>\n<p></p>", "functions": {}, "metadata": {"licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International", "description": "<p>This question tests to see if students can recognise a $p$-series and based on its $p$-value determine if it is convergent or divergent.</p>"}, "name": "Series: p series", "extensions": [], "ungrouped_variables": ["p", "num", "den", "start"], "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "tags": [], "parts": [{"scripts": {}, "prompt": "<p>This is a [[0]]&nbsp;[[1]].</p>", "gaps": [{"scripts": {}, "showFeedbackIcon": true, "displayType": "dropdownlist", "variableReplacements": [], "type": "1_n_2", "marks": 0, "variableReplacementStrategy": "originalfirst", "maxMarks": 0, "distractors": ["", ""], "choices": ["<p>convergent</p>", "<p>divergent</p>"], "minMarks": 0, "showCorrectAnswer": true, "matrix": ["if(p>1,1,0)", "if(p<=1,1,0)"], "shuffleChoices": false, "displayColumns": 0}, {"scripts": {}, "showFeedbackIcon": true, "displayType": "dropdownlist", "variableReplacements": [], "type": "1_n_2", "marks": 0, "variableReplacementStrategy": "originalfirst", "maxMarks": 0, "distractors": ["", "", "", "", "", ""], "choices": ["<p>p series</p>", "<p>geometric series</p>", "<p>arithmetic series</p>", "<p>alternating series</p>", "<p>Maclaurin series</p>", "<p>Taylor series</p>"], "minMarks": 0, "showCorrectAnswer": true, "matrix": ["1", 0, 0, "0", 0, 0], "shuffleChoices": true, "displayColumns": 0}], "showCorrectAnswer": true, "variableReplacements": [], "type": "gapfill", "showFeedbackIcon": true, "marks": 0, "variableReplacementStrategy": "originalfirst"}], "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/"}]}