// Numbas version: exam_results_page_options
{"name": "Series: divergence test", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"ungrouped_variables": ["text", "seed", "limit", "existence", "cseed", "a", "d", "coeff", "b", "c", "f", "g", "h", "alt", "cexpression", "con_to_zero", "div_to_inf", "jump_around", "con_to_non_zero", "start"], "extensions": [], "functions": {}, "rulesets": {}, "tags": [], "advice": "<p>The divergence test says that&nbsp;<strong>if</strong> the sequence of $t_k$ diverges or converges to a non-zero number<em>&nbsp;</em><strong>then</strong>&nbsp;the series $\\sum_{k=a}^\\infty t_k$ diverges. Notice it does not tell us anything about the series if the sequence of $t_k$ converges to $0$. Another way to think about this is, for a series to have any <em><g class=\"gr_ gr_62 gr-alert gr_gramm gr_inline_cards gr_run_anim Style multiReplace\" id=\"62\" data-gr-id=\"62\">chance&nbsp;</g></em><g class=\"gr_ gr_62 gr-alert gr_gramm gr_inline_cards gr_disable_anim_appear Style multiReplace\" id=\"62\" data-gr-id=\"62\"> of</g> converging then we require&nbsp;its terms to converge to zero.&nbsp;You might see the divergence test written more concisely as \\[\\lim_{k\\rightarrow\\infty} t_k\\ne 0 \\implies \\sum_{k=0}^\\infty t_k \\,\\,\\text{ diverges.}\\]</p>\n<p>a)&nbsp;</p>\n<p><span data-jme-visible=\"seed=1\">Since each term in the series is getting {text} in absolute value the sequence can never 'settle down' or 'approach' a finite number. In other words, the sequence diverges (does not converge) and so the divergence test implies that the series diverges.</span></p>\n<p><span data-jme-visible=\"seed=-1\">Since each term in the series is getting {text} in absolute value the sequence 'settles down'&nbsp;and 'approaches' zero. In other words, the sequence&nbsp;converges to zero and so the divergence test doesn't actually tell us anything about the series (it may or may not converge).</span></p>\n<p></p>\n<p>b)</p>\n<p><span data-jme-visible=\"existence=0\">Since the limit of the sequence does not exist, the sequence diverges (diverges just means does not converge) and so the divergence test tells us that the series diverges.</span></p>\n<p><span data-jme-visible=\"existence=1 and seed=-1\">Since the limit of the sequence&nbsp;is a non-zero number ($\\var[fractionNumbers]{limit}$) the divergence test&nbsp;tells us that the series diverges.</span></p>\n<p><span data-jme-visible=\"existence=1 and seed=1\">Since the limit of the sequence&nbsp;is zero, the divergence test doesn't actually tell us anything about whether the series converges or diverges.</span></p>\n<p></p>\n<p>c)</p>\n<p>To use the divergence test we need to determine whether the sequence of terms approach zero or not.&nbsp;Consider &nbsp;&nbsp;</p>\n<p style=\"text-align: center;\">{cexpression} as $k\\rightarrow \\infty$.</p>\n<p></p>\n<p></p>\n<p><span data-jme-visible=\"cseed='inf' and alt=0\">As $k$ gets larger and larger our terms get larger and larger.&nbsp;This means that&nbsp;the limit does not exist (or sometimes people prefer to say the limit equals infinity) and so the divergence test tells us that the series diverges.</span></p>\n<p></p>\n<p><span data-jme-visible=\"cseed='inf' and alt=1\">As $k$ gets larger and larger our terms get larger and larger in absolute value but alternate in sign.&nbsp;This means that&nbsp;the limit does not exist and so the divergence test tells us that the series diverges.</span></p>\n<p></p>\n<p><span data-jme-visible=\"cseed='jump'\">As $k$ gets larger and larger our terms jump around, they get positive, negative, smaller, larger... The terms do not approach a constant.&nbsp;This means that&nbsp;the limit does not exist and so the divergence test tells us that the series diverges.</span></p>\n<p></p>\n<p><span data-jme-visible=\"cseed='zero'\">As $k$ gets larger and larger our terms approach zero. The larger $k$ gets, the closer to zero the terms become. This means that the sequence converges to $0$ and so the divergence test does not tell us anything about whether the series converges or diverges.</span></p>\n<p></p>\n<p><span data-jme-visible=\"cseed='non_zero'\">As $k$ gets larger and larger our terms approach a non-zero constant. The larger $k$ gets, the closer to&nbsp;this non-zero constant the terms become. This means that the sequence converges to this non-zero constant and so the divergence test tells us that the series diverges.</span></p>\n<p></p>", "preamble": {"css": "", "js": ""}, "variable_groups": [], "name": "Series: divergence test", "parts": [{"distractors": ["", "", ""], "scripts": {}, "prompt": "<p>A series is such that each term&nbsp;has an absolute value that is&nbsp;{text} than the last. What does the <strong>divergence test</strong> tell us about this series?</p>", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "marks": 0, "shuffleChoices": false, "minMarks": 0, "displayColumns": 0, "choices": ["<p>This series converges.</p>", "<p>This series diverges.</p>", "<p>It doesn't tell us anything.</p>"], "variableReplacements": [], "showCorrectAnswer": true, "displayType": "dropdownlist", "type": "1_n_2", "matrix": [0, "if(seed=1,1,0)", "if(seed=-1,1,0)"], "maxMarks": 0}, {"distractors": ["", "", ""], "scripts": {}, "prompt": "<p>A series,&nbsp;$\\displaystyle\\sum_{k=\\var{start}}^\\infty t_k$, is such that $\\displaystyle\\lim_{k\\rightarrow \\infty}t_k$&nbsp;<span data-jme-visible=\"existence=1\">$=\\var[fractionNumbers]{limit}$</span>&nbsp;&nbsp;<span data-jme-visible=\"existence=0\">does not exist</span>. What does the <strong>divergence test</strong> tell us about this series?</p>", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "marks": 0, "shuffleChoices": false, "minMarks": 0, "displayColumns": 0, "choices": ["<p>This series converges.</p>", "<p>This series diverges.</p>", "<p>It doesn't tell us anything.</p>"], "variableReplacements": [], "showCorrectAnswer": true, "displayType": "dropdownlist", "type": "1_n_2", "matrix": [0, "if(seed=-1,1,0)", "if(seed=1,1,0)"], "maxMarks": 0}, {"distractors": ["", "", ""], "scripts": {}, "prompt": "<p>Given the series</p>\n<p style=\"text-align: center;\">$\\displaystyle\\sum_{k=\\var{start}}^\\infty$ {cexpression}</p>\n<p>What does the <strong>divergence test</strong> tell us about this series?</p>", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "marks": 0, "shuffleChoices": false, "minMarks": 0, "displayColumns": 0, "choices": ["<p>This series converges.</p>\n<p></p>", "<p>This series diverges.</p>", "<p>It doesn't tell us anything.</p>"], "variableReplacements": [], "showCorrectAnswer": true, "displayType": "dropdownlist", "type": "1_n_2", "matrix": [0, "if(cseed='inf' or cseed='jump' or cseed='non_zero',1,0)", "if(cseed='zero',1,0)"], "maxMarks": 0}], "variables": {"cseed": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "cseed", "definition": "random('zero','non_zero','inf','jump')"}, "c": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "c", "definition": "coeff[1]"}, "g": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "g", "definition": "coeff[3]"}, "b": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "b", "definition": "coeff[0]"}, "a": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "a", "definition": "random(2..12)"}, "d": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "d", "definition": "random(-12..12 except [0,a,-a])"}, "limit": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "limit", "definition": "if(seed=1,0,random(-1/9,-1/8,-1/7,-1/5,-1/4,-1/3,-1/2,-2,-1,1/9,1/8,1/7,1/5,1/4,1/3,1/2,2,1, 1/10, -1/10, -1/100, 1/100))"}, "alt": {"group": "Ungrouped variables", "templateType": "anything", "description": "<p>1 for alternating sign, 0 for not alternating sign</p>", "name": "alt", "definition": "random(0,1)"}, "start": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "start", "definition": "random(0..5)"}, "seed": {"group": "Ungrouped variables", "templateType": "anything", "description": "<p>for part b 1= inconclusive, -1=series diverges REVERSED for part a</p>", "name": "seed", "definition": "random(-1,1)"}, "jump_around": {"group": "Ungrouped variables", "templateType": "anything", "description": "<p></p>\n<p>This requires alt to equal 1...</p>\n<p>This is really 'diverge but stay bounded'.</p>", "name": "jump_around", "definition": "[\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*{d}*arctan(k)}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*{a}}\\$\",   \n\"\\$\\\\displaystyle\\\\simplify{(-1)^k}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*({a}k+{b})/({d}k+{f})}\\$\",    \n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*({a}k^2+{b}k+{c})/({d}k^2+{f}k+{g})}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*({a}k^3+{b}k^2+{c}k+{f})/({d}k^3+{f}k^2+{g}k+{h})}\\$\", \n\"\\$\\\\displaystyle\\\\simplify{sin(k)}\\$\", \n\"\\$\\\\displaystyle\\\\simplify{cos(k)}\\$\", \n\"\\$\\\\displaystyle\\\\simplify{tan(k)}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*sin(k)}\\$\", \n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*cos(k)}\\$\", \n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*tan(k)}\\$\"\n]"}, "con_to_zero": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "con_to_zero", "definition": "if(alt=1,\n[\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k/({a}^k)}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k/(k+1)}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*{a}/k!}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*k!/(k+{a})!}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*{a}^k/k!}\\$\",   \n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*({a}k+{b})/({d}k^2+{f}k+{g})}\\$\", \n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*({a}k^2+{b}k+{c})/({d}k^3+{f}k^2+{g}k+{h})}\\$\"\n],\n  [\n\"\\$\\\\displaystyle\\\\simplify{1/({a}^k)}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{1/(k+1)}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{{a}/k!}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{k!/(k+{a})!}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{{a}^k/k!}\\$\",    \n\"\\$\\\\displaystyle\\\\simplify{({a}k+{b})/({d}k^2+{f}k+{g})}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{({a}k^2+{b}k+{c})/({d}k^3+{f}k^2+{g}k+{h})}\\$\"\n])"}, "div_to_inf": {"group": "Ungrouped variables", "templateType": "anything", "description": "<p>This is really 'diverge in absolute value to infinity'</p>", "name": "div_to_inf", "definition": "if(alt=1,\n[\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*({a}^k)}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*k}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*k!/{a}}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*(k+{a})!/k!}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*k!/{a}^k}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*({d}k^2+{f}k+{g})/({a}k+{b})}\\$\", \n\"\\$\\\\displaystyle\\\\simplify{(-1)^k*({d}k^3+{f}k^2+{g}k+{h})/({a}k^2+{b}k+{c})}\\$\"\n],\n  [\n\"\\$\\\\displaystyle\\\\simplify{{a}^k}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{k}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{k!/{a}}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{(k+{a})!/k!}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{k!/{a}^k}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{({d}k^2+{f}k+{g})/({a}k+{b})}\\$\", \n\"\\$\\\\displaystyle\\\\simplify{({d}k^3+{f}k^2+{g}k+{h})/({a}k^2+{b}k+{c})}\\$\"\n])"}, "cexpression": {"group": "Ungrouped variables", "templateType": "anything", "description": "<p>cseed=random('zero','non_zero','inf','jump')</p>", "name": "cexpression", "definition": "if(cseed='zero',random(con_to_zero),\nif(cseed='non_zero',random(con_to_non_zero),\nif(cseed='inf',random(div_to_inf), \nif(cseed='jump',random(jump_around),''))))"}, "con_to_non_zero": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "con_to_non_zero", "definition": "if(alt=1,\n[\n\"\\$\\\\displaystyle\\\\simplify{({a}k^3+{b}k^2+{c}k+{f})/({d}k^3+{f}k^2+{g}k+{h})+(-1)^k/({a}^k)}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{({a}k^2+{b}k+{c})/({d}k^2+{f}k+{g})+(-1)^k/(k+1)}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{({a}k+{b})/({d}k+{f})+(-1)^k*{a}/k!}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{{a}+(-1)^k*k!/(k+{a})!}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{{d}*arctan(k)+(-1)^k*{a}^k/k!}\\$\",   \n\"\\$\\\\displaystyle\\\\simplify{({a}k+{b})/({d}k+{f})+(-1)^k*({a}k+{b})/({d}k^2+{f}k+{g})}\\$\", \n\"\\$\\\\displaystyle\\\\simplify{{a}+(-1)^k*({a}k^2+{b}k+{c})/({d}k^3+{f}k^2+{g}k+{h})}\\$\"\n],\n  [\n\"\\$\\\\displaystyle\\\\simplify{{d}*arctan(k)}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{{a}}\\$\",   \n\"\\$\\\\displaystyle\\\\simplify{({a}k+{b})/({d}k+{f})}\\$\",    \n\"\\$\\\\displaystyle\\\\simplify{({a}k^2+{b}k+{c})/({d}k^2+{f}k+{g})}\\$\",\n\"\\$\\\\displaystyle\\\\simplify{({a}k^3+{b}k^2+{c}k+{f})/({d}k^3+{f}k^2+{g}k+{h})}\\$\"\n])"}, "text": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "text", "definition": "if(seed=1,'larger', 'smaller')"}, "coeff": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "coeff", "definition": "shuffle(list(-12..12) +0+0+0+0+0+0+0+0+0+0+0+0)[0..5]"}, "f": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "f", "definition": "coeff[2]"}, "existence": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "existence", "definition": "random(0,1)"}, "h": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "h", "definition": "coeff[4]"}}, "metadata": {"description": "<p>Test whether a student knows the divergence test of a series, and how to use it. Series include those that the <g class=\"gr_ gr_7 gr-alert gr_spell gr_inline_cards gr_run_anim ContextualSpelling ins-del multiReplace\" id=\"7\" data-gr-id=\"7\">diverence</g> test is inconclusive for. This question could be better in that it could go through the working of determining the limit but I&nbsp;hope to&nbsp;make a separate question which deals with that.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "variablesTest": {"maxRuns": 100, "condition": ""}, "statement": "<p>This question is about the divergence test for series.</p>", "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/"}]}