// Numbas version: exam_results_page_options {"name": "Series: ratio test", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"ungrouped_variables": ["seed", "percent", "limit", "a", "d", "coeff", "b", "c", "f", "g", "h", "alt", "cseed", "cexpression", "ratio_con_to_zero", "ratio_con_to_less_than_one", "ratio_con_to_one", "ratio_con_to_big", "ratio_div_to_inf", "start"], "extensions": [], "functions": {}, "rulesets": {}, "tags": [], "advice": "

The ratio test says, given a series $\\sum_{k=a}^\\infty t_k$

if $\\lim_{k\\rightarrow\\infty}\\left|\\frac{t_{k+1}}{t_k}\\right|<1$, then the series converges
if $\\lim_{k\\rightarrow\\infty}\\left|\\frac{t_{k+1}}{t_k}\\right|>1$, then the series diverges
if $\\left|\\frac{t_{k+1}}{t_k}\\right|\\rightarrow\\infty$ as $k\\rightarrow \\infty$, then the series diverges

Notice, if the limit doesn't exist for some other reason or is equal to $1$ the test doesn't tell us anything, some might say the test is 'inconclusive' or 'fails'.

Since $\\lim_{k\\rightarrow\\infty}\\left|\\frac{t_{k+1}}{t_k}\\right|=\\frac{\\var{percent}}{100}<1$ the ratio test tells us that the series converges.

Since $\\lim_{k\\rightarrow\\infty}\\left|\\frac{t_{k+1}}{t_k}\\right|=\\frac{\\var{percent}}{100}>1$ the ratio test tells us that the series diverges.

Since $\\lim_{k\\rightarrow\\infty}\\left|\\frac{t_{k+1}}{t_k}\\right|=1$ the ratio test doesn't tell us anything.

Since $\\lim_{k\\rightarrow\\infty}\\left|\\frac{t_{k+1}}{t_k}\\right|=\\var[fractionNumbers]{limit}<1$ the ratio test tells us that the series converges.

Since $\\lim_{k\\rightarrow\\infty}\\left|\\frac{t_{k+1}}{t_k}\\right|=\\var[fractionNumbers]{limit}>1$ the ratio test tells us that the series diverges.

Since $\\lim_{k\\rightarrow\\infty}\\left|\\frac{t_{k+1}}{t_k}\\right|=1$ the ratio test doesn't tell us anything.

To use the ratio test we need to determine the value that the absolute value of the ratio of consecutive terms converge to.

Given $t_k=${cexpression} we use algebra and limit laws to determine

\\[\\lim_{k\\rightarrow\\infty}\\frac{t_{k+1}}{t_k}=0\\]

\\[\\lim_{k\\rightarrow\\infty}\\frac{t_{k+1}}{t_k}=\\simplify[fractionNumbers]{{1/a}}\\]

\\[\\lim_{k\\rightarrow\\infty}\\frac{t_{k+1}}{t_k}=1\\]

\\[\\lim_{k\\rightarrow\\infty}\\frac{t_{k+1}}{t_k}=\\var{a}\\]

and therefore by the ratio test the series converges.

and therefore by the ratio test the series diverges.

and therefore the ratio test doesn't tell us anything.

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A series is such that as we get further and further along the sequence, each term has an absolute value that approaches $\\var{percent}\\%$ of the absolute value of the previous term. What does the ratio test tell us about this series?

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This series converges.

", "

This series diverges.

", "

It doesn't tell us anything.

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A series, $\\displaystyle\\sum_{k=\\var{start}}^\\infty t_k$, is such that $\\displaystyle\\lim_{k\\rightarrow \\infty}\\left\\vert\\frac{t_{k+1}}{t_k}\\right\\vert $ $=\\var[fractionNumbers]{limit}$. What does the ratio test tell us about this series?

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This series converges.

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This series diverges.

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It doesn't tell us anything.

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Given the series

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

What does the ratio test tell us about this series?

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This series converges.

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This series diverges.

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It doesn't tell us anything.

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1 for alternating sign, 0 for not alternating sign

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I have omitted these but you might like to use them, it depends on what your course notes say about the ratio test...

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random(0,0.5,1,2)

", "name": "cexpression", "definition": "if(cseed=0,random(ratio_con_to_zero),\nif(cseed=0.5,random(ratio_con_to_less_than_one),\nif(cseed=1,random(ratio_con_to_one),\n random(ratio_con_to_big))))\n"}, "percent": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "percent", "definition": "if(seed=-1,random(1..99), if(seed=0,100, random(101..200)))"}, "ratio_con_to_one": {"group": "Ungrouped variables", "templateType": "anything", "description": "

These have ratios that converge to 1 and so the ratio test is inconclusive

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Test whether a student knows the ratio test of a series, and how to use it. Series include those that the ratio 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 ratio 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/"}]}