// Numbas version: finer_feedback_settings
{"name": "alternating series 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "alternating series 1", "tags": [], "metadata": {"description": "<p>alternating series&nbsp;</p>", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "<p>Decide whether the following series is convergent or not</p>", "advice": "<p>If you view the series as&nbsp;</p>\n<p>\\[\\frac{1}{\\sqrt[\\var{m}]{2}-1} - \\frac{1}{\\sqrt[\\var{m}]{2}+1} + \\frac{1}{\\sqrt[\\var{m}]{3}-1} - \\frac{1}{\\sqrt[\\var{m}]{3}+1} + \\frac{1}{\\sqrt[\\var{m}]{4}-1} - \\frac{1}{\\sqrt[\\var{m}]{4}+1} + \\cdots\\]</p>\n<p>It is an alternating series. However, alternating series does not work (why not?)</p>\n<p></p>\n<p>Instead, observe that&nbsp;</p>\n<p>\\[\\frac{1}{\\sqrt[\\var{m}]{n}-1} - \\frac{1}{\\sqrt[\\var{m}]{n}+1} = \\frac{1}{n^{2/{\\var{m}}}-1}.\\]</p>\n<p>Also, observe that&nbsp;</p>\n<p>\\[ \\frac{1}{n^{2/{\\var{m}}}} \\leq \\frac{1}{n^{2/{\\var{m}}}-1}. \\]</p>\n<p>By $p$-series test, $\\displaystyle \\sum_{n=2}^\\infty \\frac{1}{n^{2/{\\var{m}}}}$ is divergent (as $2/{\\var{m}} \\leq 1$). Then by comparison test, $\\displaystyle \\sum_{n=2}^\\infty \\frac{1}{n^{2/{\\var{m}}}-1} = \\sum_{n=2}^\\infty \\left(\\frac{1}{\\sqrt[\\var{m}]{n}-1} - \\frac{1}{\\sqrt[\\var{m}]{n}+1}\\right)$ is also divergent.&nbsp;</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"m": {"name": "m", "group": "Ungrouped variables", "definition": "random(list(2..9))", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["m"], "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>$\\displaystyle\\sum_{n=2}^\\infty \\left( \\frac{1}{\\sqrt[\\var{m}]{n}-1} - \\frac{1}{\\sqrt[\\var{m}]{n}+1}\\right)$</p>", "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["converges", "diverges"], "matrix": ["0", "1"], "distractors": ["", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question", "contributors": [{"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18261/"}], "resources": []}]}], "contributors": [{"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18261/"}]}