// Numbas version: exam_results_page_options {"name": "Series: The root test", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"tags": [], "name": "Series: The root test", "advice": "

The root test says

\n
\n
•  If $\\lim_{n\\rightarrow\\infty}\\sqrt[n]{\\vert a_n\\vert}<1$, then the series $\\sum_{n=1}^\\infty a_n$ is convergent (and absolutely convergent).
• \n
• If $\\lim_{n\\rightarrow\\infty}\\sqrt[n]{\\vert a_n\\vert}>1$ or $\\sqrt[n]{\\vert a_n\\vert}$ diverges to $\\infty$, then the series $\\sum_{n=1}^\\infty a_n$ is divergent.
• \n
\n

Notice, if $\\lim_{n\\rightarrow\\infty}\\sqrt[n]{\\vert a_n\\vert}=1$ or if $\\sqrt[n]{\\vert a_n\\vert}$ diverges in some other way, then the test doesn't say anything, in these cases some might say the test is 'inconclusive' or 'fails'.

\n

\n

a)

\n

Since $\\lim_{n\\rightarrow\\infty}\\sqrt[n]{\\vert a_n\\vert}=\\var[fractionNumbers]{alimit}<1$ the series converges by the root test. Since $\\lim_{n\\rightarrow\\infty}\\sqrt[n]{\\vert a_n\\vert}=\\var[fractionNumbers]{alimit}>1$ the series diverges by the root test. Since $\\lim_{n\\rightarrow\\infty}\\sqrt[n]{\\vert a_n\\vert}=\\var[fractionNumbers]{alimit}$ the root test doesn't tell us anything.

\n

\n

b)

\n

Since $\\lim_{n\\rightarrow\\infty}\\sqrt[n]{\\vert a_n\\vert}=\\var[fractionNumbers]{blimit}<1$ the series converges by the root test. Since $\\lim_{n\\rightarrow\\infty}\\sqrt[n]{\\vert a_n\\vert}=\\var[fractionNumbers]{blimit}>1$ the series diverges by the root test. Since $\\lim_{n\\rightarrow\\infty}\\sqrt[n]{\\vert a_n\\vert}=\\var[fractionNumbers]{blimit}$ the root test doesn't tell us anything.

\n

\n

c)

\n

For $\\displaystyle\\sum_{k=\\var{start}}^\\infty$ {cexpression}, we can determine that \$\\lim_{k\\rightarrow\\infty}\\sqrt[k]{\\vert a_k\\vert }=\\var[fractionNumbers]{cexpression}\$

\n

and so the series converges. and so the series diverges. and so the test is inconclusive.

\n

\n

To actually determine these limits you might need to use the handy fact that $\\lim_{n\\rightarrow\\infty} n^{1/n}=1$. Looking at why this is true will also give insight into how to calculate some of these limits (notice the nice little 'trick' of applying $e^{\\ln()}$, using log laws and limit laws)

\n

\\\begin{align}\\lim_{n\\rightarrow\\infty} n^{1/n}&=\\lim_{n\\rightarrow\\infty} e^{\\ln(n^{1/n})}\\\\&=\\lim_{n\\rightarrow\\infty} e^{\\frac{1}{n}\\ln(n)}\\\\&= e^{\\lim_{n\\rightarrow\\infty}\\frac{1}{n}\\ln(n)}\\\\&= e^{\\lim_{n\\rightarrow\\infty}\\frac{\\ln(n)}{n}}\\end{align}\

\n

but this limit is an indeterminate form so we apply L'Hopital's rule* and instead take the limit of the derivative of the numerator over the derivative of the denominator

\n

\\\begin{align}e^{\\lim_{n\\rightarrow\\infty}\\frac{\\ln(n)}{n}}&=e^{\\lim_{n\\rightarrow\\infty}\\frac{\\frac{d}{dn}(\\ln(n))}{\\frac{d}{dn}(n)}}\\\\&=e^{\\lim_{n\\rightarrow\\infty}\\frac{1}{n}}\\\\&=e^0\\\\&=1\\end{align}\

\n

\n

Also, notice that we also have the related handy fact that $\\lim_{n\\rightarrow\\infty} (an+b)^{1/n}=1$ by nearly the same proof.

\n

\n

*If you haven't learnt L'Hopital's rule, just notice that $n$ grows a lot faster than $\\ln(n)$ as $n\\rightarrow\\infty$ and so it will dominate.

\n

\n

\n

", "variable_groups": [], "preamble": {"css": "", "js": ""}, "metadata": {"description": "

Test whether a student knows the root test of a series, and how to use it. Series include those that the root 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"}, "functions": {}, "variablesTest": {"maxRuns": 100, "condition": ""}, "statement": "

This question is about the root test for series.

", "extensions": [], "parts": [{"variableReplacementStrategy": "originalfirst", "type": "1_n_2", "marks": 0, "maxMarks": 0, "showCorrectAnswer": true, "matrix": "aseed", "scripts": {}, "showFeedbackIcon": true, "minMarks": 0, "shuffleChoices": false, "prompt": "

A series is such that as $n$ gets larger and larger the $n$th root of the absolute value of the $n$th term approaches $\\var[fractionNumbers]{alimit}$. What does the root test tell us about this series?

", "displayColumns": 0, "choices": ["

This series converges.

", "

This series diverges.

", "

It doesn't tell us anything.

"], "variableReplacements": [], "displayType": "dropdownlist"}, {"variableReplacementStrategy": "originalfirst", "type": "1_n_2", "marks": 0, "maxMarks": 0, "showCorrectAnswer": true, "matrix": "bseed", "scripts": {}, "showFeedbackIcon": true, "minMarks": 0, "shuffleChoices": false, "prompt": "

A series, $\\displaystyle\\sum_{n=\\var{start}}^\\infty a_n$, is such that $\\displaystyle\\lim_{n\\rightarrow \\infty}\\sqrt[n]{\\vert a_n\\vert}=\\var[fractionNumbers]{blimit}$. What does the root test tell us about this series?

", "displayColumns": 0, "choices": ["

This series converges.

", "

This series diverges.

", "

It doesn't tell us anything.

"], "variableReplacements": [], "displayType": "dropdownlist"}, {"variableReplacementStrategy": "originalfirst", "type": "1_n_2", "marks": 0, "maxMarks": 0, "showCorrectAnswer": true, "matrix": "cseed", "scripts": {}, "showFeedbackIcon": true, "minMarks": 0, "shuffleChoices": false, "prompt": "

Given the series

\n

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

\n

", "displayColumns": 0, "choices": ["

This series converges.

", "

This series diverges.

", "

It doesn't tell us anything.

"], "variableReplacements": [], "displayType": "dropdownlist"}], "rulesets": {}, "variables": {"blimit": {"group": "Ungrouped variables", "definition": "switch(bseed=[1,0,0],random(1/10,1/5,3/10,2/5,1/2,3/5,7/10,4/5,9/10,1/3,1/4,2/3,3/4),bseed=[0,1,0],random(10/3,5/2,5/3,10/7,5/4,10/9,3/2,4/3, 2, 3, 4, 5, 6, 7, 8, 9, 10), 1)", "templateType": "anything", "name": "blimit", "description": ""}, "d": {"group": "Ungrouped variables", "definition": "num", "templateType": "anything", "name": "d", "description": ""}, "g": {"group": "Ungrouped variables", "definition": "num", "templateType": "anything", "name": "g", "description": ""}, "start": {"group": "Ungrouped variables", "definition": "random(1..6)", "templateType": "anything", "name": "start", "description": ""}, "b": {"group": "Ungrouped variables", "definition": "num", "templateType": "anything", "name": "b", "description": ""}, "bseed": {"group": "Ungrouped variables", "definition": "random([[1,0,0],[0,1,0],[0,0,1]] except [aseed])", "templateType": "anything", "name": "bseed", "description": ""}, "num": {"group": "Ungrouped variables", "definition": "shuffle(2..12)", "templateType": "anything", "name": "num", "description": ""}, "aseed": {"group": "Ungrouped variables", "definition": "random([1,0,0],[0,1,0],[0,0,1])", "templateType": "anything", "name": "aseed", "description": "

converge, diverge, inconclusive

"}, "diverge": {"group": "Ungrouped variables", "definition": "[\n [\"\\$\\\\displaystyle\\\\simplify{(-{a})^k/({b}k+{c})}\\$\",a],\n [\"\\$\\\\displaystyle\\\\simplify{({a})^k/({b}k+{c})}\\$\",a],\n [\"\\$\\\\displaystyle\\\\simplify{(-{a})^k/({b}k^{d}+{c})}\\$\",a],\n [\"\\$\\\\displaystyle\\\\simplify{({a})^k/({b}k^{d}+{c})}\\$\",a],\n [\"\\$\\\\displaystyle\\\\simplify{(-{a})^k/({b}k^3+{c}k^2+{d}k+{f})}\\$\",a],\n [\"\\$\\\\displaystyle\\\\simplify{({a})^k/({b}k^3+{c}k^2+{d}k+{f})}\\$\",a],\n \n [\"\\$\\\\displaystyle\\\\simplify{(k)^k/({b}k^3+{c}k^2+{d}k+{f})}\\$\",infinity],\n [\"\\$\\\\displaystyle\\\\simplify{(k)^k/({a}^({b}+{c}k))}\\$\",infinity],\n \n [\"\\$\\\\displaystyle\\\\simplify{(({a+c}k+{d})/({a}k+{b}))^k}\\$\",(a+c)/a],\n [\"\\$\\\\displaystyle\\\\simplify{(({a+c}k^2+{d}k+{g})/({a}k^2+{b}k+{f}))^k}\\$\",(a+c)/a],\n [\"\\$\\\\displaystyle\\\\simplify{(({a+c}k^3+{d}k+{g})/({a}k^3+{b}k^2+{f}))^k}\\$\",(a+c)/a],\n [\"\\$\\\\displaystyle\\\\simplify{(-({a+c}k+{d})/({a}k+{b}))^k}\\$\",(a+c)/a],\n [\"\\$\\\\displaystyle\\\\simplify{(-({a+c}k^2+{d}k+{g})/({a}k^2+{b}k+{f}))^k}\\$\",(a+c)/a],\n [\"\\$\\\\displaystyle\\\\simplify{(-({a+c}k^3+{d}k+{g})/({a}k^3+{b}k^2+{f}))^k}\\$\",(a+c)/a],\n \n [\"\\$\\\\displaystyle\\\\simplify{({a+b}k^k+{c}k^(k-{d}))/({a}(k)^k)}\\$\",(a+b)/a],\n [\"\\$\\\\displaystyle\\\\simplify{-({a+b}k^k+{c}k^(k-{d}))/({a}(k)^k)}\\$\",(a+b)/a],\n [\"\\$\\\\displaystyle\\\\simplify{({a+b}k^k-{c}k^(k-{d}))/({a}(k)^k)}\\$\",(a+b)/a]\n]\n\n", "templateType": "anything", "name": "diverge", "description": ""}, "cseed": {"group": "Ungrouped variables", "definition": "random([1,0,0],[0,1,0],[0,0,1])", "templateType": "anything", "name": "cseed", "description": ""}, "alimit": {"group": "Ungrouped variables", "definition": "switch(aseed=[1,0,0],random(1/10,1/5,3/10,2/5,1/2,3/5,7/10,4/5,9/10,1/3,1/4,2/3,3/4),aseed=[0,1,0],random(10/3,5/2,5/3,10/7,5/4,10/9,3/2,4/3, 2, 3, 4, 5, 6, 7, 8, 9, 10,infinity), 1)", "templateType": "anything", "name": "alimit", "description": ""}, "a": {"group": "Ungrouped variables", "definition": "num", "templateType": "anything", "name": "a", "description": ""}, "c": {"group": "Ungrouped variables", "definition": "num", "templateType": "anything", "name": "c", "description": ""}, "converge": {"group": "Ungrouped variables", "definition": "[\n [\"\\$\\\\displaystyle\\\\simplify{(({a}k+{b})/({a+c}k+{d}))^k}\\$\",a/(a+c)],\n [\"\\$\\\\displaystyle\\\\simplify{(({a}k^2+{b}k+{f})/({a+c}k^2+{d}k+{g}))^k}\\$\",a/(a+c)],\n [\"\\$\\\\displaystyle\\\\simplify{(({a}k^3+{b}k^2+{f})/({a+c}k^3+{d}k+{g}))^k}\\$\",a/(a+c)],\n [\"\\$\\\\displaystyle\\\\simplify{(-({a}k+{b})/({a+c}k+{d}))^k}\\$\",a/(a+c)],\n [\"\\$\\\\displaystyle\\\\simplify{(-({a}k^2+{b}k+{f})/({a+c}k^2+{d}k+{g}))^k}\\$\",a/(a+c)],\n [\"\\$\\\\displaystyle\\\\simplify{(-({a}k^3+{b}k^2+{f})/({a+c}k^3+{d}k+{g}))^k}\\$\",a/(a+c)],\n \n [\"\\$\\\\displaystyle\\\\simplify{{a}(k)^k/({a+b}k^k+{c}k^(k-{d}))}\\$\",a/(a+b)],\n [\"\\$\\\\displaystyle\\\\simplify{-{a}(k)^k/({a+b}k^k+{c}k^(k-{d}))}\\$\",a/(a+b)],\n [\"\\$\\\\displaystyle\\\\simplify{{a}(k)^k/({a+b}k^k-{c}k^(k-{d}))}\\$\",a/(a+b)]\n \n [\"\\$\\\\displaystyle\\\\simplify{({b}k^3+{c}k^2+{d}k+{f})/(k)^k}\\$\",0],\n [\"\\$\\\\displaystyle\\\\simplify{({a}^({b}+{c}k))/(k)^k}\\$\",0],\n \n [\"\\$\\\\displaystyle\\\\simplify{(-{a})^(-k)/({b}k+{c})}\\$\",1/a],\n [\"\\$\\\\displaystyle\\\\simplify{({a})^(-k)/({b}k+{c})}\\$\",1/a],\n [\"\\$\\\\displaystyle\\\\simplify{(-{a})^(-k)/({b}k^{d}+{c})}\\$\",1/a],\n [\"\\$\\\\displaystyle\\\\simplify{({a})^(-k)/({b}k^{d}+{c})}\\$\",1/a],\n [\"\\$\\\\displaystyle\\\\simplify{(-{a})^(-k)/({b}k^3+{c}k^2+{d}k+{f})}\\$\",1/a],\n [\"\\$\\\\displaystyle\\\\simplify{({a})^(-k)/({b}k^3+{c}k^2+{d}k+{f})}\\$\",1/a],\n \n ]", "templateType": "anything", "name": "converge", "description": ""}, "f": {"group": "Ungrouped variables", "definition": "num", "templateType": "anything", "name": "f", "description": ""}, "inconclusive": {"group": "Ungrouped variables", "definition": "[\n [\"\\$\\\\displaystyle\\\\simplify{(-1)^k/({b}k+{c})}\\$\",1],\n [\"\\$\\\\displaystyle\\\\simplify{(1/({b}k+{c})}\\$\",1],\n [\"\\$\\\\displaystyle\\\\simplify{(-1)^k/({b}k^{d}+{c})}\\$\",1],\n [\"\\$\\\\displaystyle\\\\simplify{(1/({b}k^{d}+{c})}\\$\",1],\n [\"\\$\\\\displaystyle\\\\simplify{(-1)^k/({b}k^3+{c}k^2+{d}k+{f})}\\$\",1],\n [\"\\$\\\\displaystyle\\\\simplify{1/({b}k^3+{c}k^2+{d}k+{f})}\\$\",1],\n [\"\\$\\\\displaystyle\\\\simplify{(k)^k/(k^k+{c}k^(k-{d}))}\\$\",1],\n [\"\\$\\\\displaystyle\\\\simplify{-(k)^k/(k^k+{c}k^(k-{d}))}\\$\",1],\n [\"\\$\\\\displaystyle\\\\simplify{(k)^k/(k^k-{c}k^(k-{d}))}\\$\",1],\n \n [\"\\$\\\\displaystyle\\\\simplify{(({a}k+{b})/({a}k+{d}))^k}\\$\",1],\n [\"\\$\\\\displaystyle\\\\simplify{(({a}k^2+{b}k+{f})/({a}k^2+{d}k+{g}))^k}\\$\",1],\n [\"\\$\\\\displaystyle\\\\simplify{(({a}k^3+{b}k^2+{f})/({a}k^3+{d}k+{g}))^k}\\$\",1],\n [\"\\$\\\\displaystyle\\\\simplify{(-({a}k+{b})/({a}k+{d}))^k}\\$\",1],\n [\"\\$\\\\displaystyle\\\\simplify{(-({a}k^2+{b}k+{f})/({a}k^2+{d}k+{g}))^k}\\$\",1],\n [\"\\$\\\\displaystyle\\\\simplify{(-({a}k^3+{b}k^2+{f})/({a}k^3+{d}k+{g}))^k}\\$\",1],\n \n [\"\\$\\\\displaystyle\\\\simplify{{a}(k)^k/({a}k^k+{c}k^(k-{d}))}\\$\",1],\n [\"\\$\\\\displaystyle\\\\simplify{-{a}(k)^k/({a}k^k+{c}k^(k-{d}))}\\$\",1],\n [\"\\$\\\\displaystyle\\\\simplify{{a}(k)^k/({a}k^k-{c}k^(k-{d}))}\\$\",1]\n \n ]", "templateType": "anything", "name": "inconclusive", "description": ""}, "cexpression": {"group": "Ungrouped variables", "definition": "switch(cseed=[1,0,0],\nconverge,\ncseed=[0,1,0],\ndiverge,\ninconclusive)", "templateType": "anything", "name": "cexpression", "description": ""}}, "ungrouped_variables": ["aseed", "alimit", "bseed", "blimit", "start", "cseed", "num", "a", "b", "c", "d", "f", "g", "cexpression", "converge", "diverge", "inconclusive"], "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/"}]}