// Numbas version: exam_results_page_options {"name": "Series: divergence test", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": 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": "

The divergence test says that if the sequence of $t_k$ diverges or converges to a non-zero number then 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 chance  of converging then we require its terms to converge to zero. 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.}\$

\n

a)

\n

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.

\n

Since each term in the series is getting {text} in absolute value the sequence 'settles down' and 'approaches' zero. In other words, the sequence converges to zero and so the divergence test doesn't actually tell us anything about the series (it may or may not converge).

\n

\n

b)

\n

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.

\n

Since the limit of the sequence is a non-zero number ($\\var[fractionNumbers]{limit}$) the divergence test tells us that the series diverges.

\n

Since the limit of the sequence is zero, the divergence test doesn't actually tell us anything about whether the series converges or diverges.

\n

\n

c)

\n

To use the divergence test we need to determine whether the sequence of terms approach zero or not. Consider

\n

{cexpression} as $k\\rightarrow \\infty$.

\n

\n

\n

As $k$ gets larger and larger our terms get larger and larger. This means that 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.

\n

\n

As $k$ gets larger and larger our terms get larger and larger in absolute value but alternate in sign. This means that the limit does not exist and so the divergence test tells us that the series diverges.

\n

\n

As $k$ gets larger and larger our terms jump around, they get positive, negative, smaller, larger... The terms do not approach a constant. This means that the limit does not exist and so the divergence test tells us that the series diverges.

\n

\n

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.

\n

\n

As $k$ gets larger and larger our terms approach a non-zero constant. The larger $k$ gets, the closer to 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.

\n

", "preamble": {"css": "", "js": ""}, "variable_groups": [], "name": "Series: divergence test", "parts": [{"distractors": ["", "", ""], "scripts": {}, "prompt": "

A series is such that each term has an absolute value that is {text} than the last. What does the divergence test tell us about this series?

", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "marks": 0, "shuffleChoices": false, "minMarks": 0, "displayColumns": 0, "choices": ["

This series converges.

", "

This series diverges.

", "

It doesn't tell us anything.

"], "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": "

A series, $\\displaystyle\\sum_{k=\\var{start}}^\\infty t_k$, is such that $\\displaystyle\\lim_{k\\rightarrow \\infty}t_k$ $=\\var[fractionNumbers]{limit}$  does not exist. What does the divergence test tell us about this series?

", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "marks": 0, "shuffleChoices": false, "minMarks": 0, "displayColumns": 0, "choices": ["

This series converges.

", "

This series diverges.

", "

It doesn't tell us anything.

"], "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": "

Given the series

\n

$\\displaystyle\\sum_{k=\\var{start}}^\\infty$ {cexpression}

\n

", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "marks": 0, "shuffleChoices": false, "minMarks": 0, "displayColumns": 0, "choices": ["

This series converges.

\n

", "

This series diverges.

", "

It doesn't tell us anything.

"], "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": "

1 for alternating sign, 0 for not alternating sign

", "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": "

for part b 1= inconclusive, -1=series diverges REVERSED for part a

", "name": "seed", "definition": "random(-1,1)"}, "jump_around": {"group": "Ungrouped variables", "templateType": "anything", "description": "

\n

This requires alt to equal 1...

\n

This is really 'diverge but stay bounded'.

", "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": "

This is really 'diverge in absolute value to infinity'

", "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": "

cseed=random('zero','non_zero','inf','jump')

", "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": "

Test whether a student knows the divergence test of a series, and how to use it. Series include those that the diverence test is inconclusive for. This question could be better in that it could go through the working of determining the limit but I hope to make a separate question which deals with that.

", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "variablesTest": {"maxRuns": 100, "condition": ""}, "statement": "

This question is about the divergence test for series.

", "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/"}]}