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{"name": "ratio test - medium", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "ratio test - medium", "tags": [], "metadata": {"description": "<p>ratio test question</p>", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "<p>Decide whether the following series converges:</p>", "advice": "<p><strong>a)</strong> Apply ratio test&nbsp;</p>\n<p>\\[\\displaystyle\\lim_{n\\to \\infty} \\frac{\\frac{(n+1)^2+2(n+1)+1}{3^{n+1} + 2}}{\\frac{n^2+2n+1}{3^{n} + 2}}&nbsp; =<br/>\\lim_{n\\to \\infty}\\left( \\frac{(n+1)^2+2(n+1)+1}{3^{n+1} + 2}\\cdot\\frac{ 3^{n} + 2}{n^2+2n+1}\\right)&nbsp;<br/>= \\lim_{n\\to \\infty}\\left( \\frac{(n+1)^2+2(n+1)+1}{n^2+2n+1}\\cdot\\frac{ 3^{n} + 2}{3^{n+1 } + 2}\\right)<br/>&nbsp;=\\frac{1}{3}.\\]</p>\n<p>Since the limit is less than $1$, the series is convergent.</p>\n<p></p>\n<p><strong>b)</strong></p>\n<p>Apply ratio test&nbsp;</p>\n<p>\\[\\displaystyle\\lim_{n\\to \\infty} \\frac{\\frac{(n+1)^2+2(n+1)+1}{(n+1)!}}{\\frac{n^2+2n+1}{n!}}&nbsp; =<br/>\\lim_{n\\to \\infty}\\left( \\frac{(n+1)^2+2(n+1)+1}{(n+1)!}\\cdot\\frac{ n!}{n^2+2n+1}\\right)&nbsp;<br/>= \\lim_{n\\to \\infty}\\left( \\frac{(n+1)^2+2(n+1)+1}{n^2+2n+1}\\cdot\\frac{ n!}{(n+1)!}\\right)<br/>&nbsp;=\\lim_{n\\to \\infty}\\frac{1}{n} = 0.\\]</p>\n<p>Since the limit is less than $1$, the series is convergent.</p>\n<p></p>\n<p><strong>c)</strong></p>\n<p>Apply ratio test&nbsp;</p>\n<p>\\[\\displaystyle\\lim_{n\\to \\infty} \\frac{\\frac{(2(n+1)+1)}{5^{n+1}((n+1)!)^2}}{\\frac{(2n+1)}{5^{n}(n!)^2}}&nbsp; =<br/>\\lim_{n\\to \\infty}\\left( \\frac{(2(n+1)+1)}{5^{n+1}((n+1)!)^2}\\cdot\\frac{5^{n}(n!)^2}{2n+1 }\\right)&nbsp;<br/>= \\lim_{n\\to \\infty}\\left( \\frac{2(n+1)+1}{2n+1}\\cdot\\frac{ 5^{n}(n!)^2}{5^{n+1}((n+1)!)^2}\\right)<br/>&nbsp;=\\lim_{n\\to \\infty}\\frac{1}{5n^2} = 0.\\]</p>\n<p>Since the limit is less than $1$, the series is convergent.</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\displaystyle\\sum_{n=1}^\\infty \\frac{n^2+2n+1}{3^n +2}$&nbsp;</p>", "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["converges", "diverges"], "matrix": ["1", 0], "distractors": ["", ""]}, {"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\displaystyle\\sum_{n=1}^\\infty \\frac{n^2+2n+1}{n!}$&nbsp;</p>", "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["converges", "diverges"], "matrix": ["1", 0], "distractors": ["", ""]}, {"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\displaystyle\\sum_{n=1}^\\infty \\frac{(2n+1)}{5^n (n!)^2}$&nbsp;</p>", "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["converges", "diverges"], "matrix": ["1", 0], "distractors": ["", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18261/"}]}]}], "contributors": [{"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18261/"}]}