// Numbas version: exam_results_page_options
{"name": "Monotonic sequence theorem", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"metadata": {"description": "<p>Test whether a student knows the&nbsp;monotonic sequence theorem.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>The monotonic sequence theorem is&nbsp;\"Every bounded, monotonic sequence is convergent\". Which can be broken up into two cases:</p>\n<p>&bull; If $\\{a_n\\}$ is monotonically non-decreasing (i.e. $a_{n} \\le a_{n+1}$ for all $n$) and bounded above (i.e. there exists a number $M$ such that $a_n\\le M$ for all $n$) then the sequence converges (i.e. there exists an $L$ such that as $n\\rightarrow \\infty$, $a_n\\rightarrow L$)&nbsp;</p>\n<p>&bull; If $\\{a_n\\}$ is monotonically non-increasing (i.e. $a_{n} \\ge a_{n+1}$ for all $n$) and bounded&nbsp;below (i.e. there exists a number $m$ such that $a_n\\ge m$ for all $n$) then the sequence converges&nbsp;(i.e. there exists an $L$ such that as $n\\rightarrow \\infty$, $a_n\\rightarrow L$)</p>\n<p>Also, a sequence is called 'bounded' if it is both bounded above and bounded below.</p>\n<p></p>\n<p>Notice, if $\\{a_n\\}$ is monotonically non-increasing and bounded above, or if $\\{a_n\\}$ is monotonically non-decreasing and bounded below, the theorem isn't useful since the sequence could still diverge to $-\\infty$ or $\\infty$ (respectively) or converge.&nbsp;</p>\n<p></p>\n<p>a)</p>\n<p><span data-jme-visible=\"aseed&lt;=3\">By the monotonic sequence theorem, the sequence converges.</span></p>\n<p><span data-jme-visible=\"aseed&gt;=4\">The monotonic sequence theorem is not applicable here.</span></p>\n<p></p>\n<p>b)</p>\n<p><span data-jme-visible=\"bdict[&quot;M&quot;]=&quot;MND&quot; and bdict[&quot;B&quot;]=&quot;B&quot;\">The sequence&nbsp;$a_n=${bdict[\"a_n\"]} for $n\\ge1$ is monotonically non-decreasing (notice its derivative is positive for all $n\\ge 1$) and is bounded (i.e. bounded above and below).&nbsp;&nbsp;</span> <span data-jme-visible=\"bdict[&quot;M&quot;]=&quot;MNI&quot; and bdict[&quot;B&quot;]=&quot;B&quot;\">The sequence&nbsp;$a_n=${bdict[\"a_n\"]} for $n\\ge1$ is monotonically non-increasing (notice its derivative is negative for all $n\\ge 1$) and is bounded (i.e. bounded above and below).&nbsp;&nbsp;</span> <span data-jme-visible=\"bdict[&quot;M&quot;]=&quot;MND&quot; and bdict[&quot;B&quot;]=&quot;NB&quot;\">The sequence&nbsp;$a_n=${bdict[\"a_n\"]} for $n\\ge1$ is monotonically non-decreasing (notice its derivative is positive for all $n\\ge 1$)&nbsp;but is not bounded.&nbsp;&nbsp;</span> <span data-jme-visible=\"bdict[&quot;M&quot;]=&quot;MNI&quot; and bdict[&quot;B&quot;]=&quot;NB&quot;\">The sequence&nbsp;$a_n=${bdict[\"a_n\"]} for $n\\ge1$ is monotonically non-increasing (notice its derivative is&nbsp;negative for all $n\\ge 1$)&nbsp;but is not bounded.&nbsp;&nbsp;</span> <span data-jme-visible=\"bdict[&quot;M&quot;]=&quot;NM&quot; and bdict[&quot;B&quot;]=&quot;NB&quot;\">The sequence&nbsp;$a_n=${bdict[\"a_n\"]} for $n\\ge1$ is not monotonic (notice it oscillates) and is not bounded.&nbsp;&nbsp;</span> <span data-jme-visible=\"bdict[&quot;M&quot;]=&quot;NM&quot; and bdict[&quot;B&quot;]=&quot;B&quot;\">The sequence&nbsp;$a_n=${bdict[\"a_n\"]} for $n\\ge1$ is not monotonic (notice it oscillates)&nbsp;but is bounded&nbsp;(i.e. bounded above and below).</span>&nbsp;&nbsp;</p>\n<p><span data-jme-visible=\"bdict[&quot;MCT&quot;]=&quot;C&quot;\">Therefore, by the monotonic sequence&nbsp;theorem, this sequence converges.</span> <span data-jme-visible=\"bdict[&quot;MCT&quot;]=&quot;NA&quot;\">Therefore the monotonic sequence theorem is not applicable to this sequence.</span></p>", "ungrouped_variables": ["aseed", "coeff", "a", "b", "c", "d", "bdictlist", "bdict"], "statement": "<p>This question is about the monotonic sequence theorem.</p>", "extensions": [], "rulesets": {}, "name": "Monotonic sequence theorem", "functions": {}, "parts": [{"scripts": {}, "displayColumns": 0, "marks": 0, "shuffleChoices": false, "variableReplacements": [], "maxMarks": 0, "variableReplacementStrategy": "originalfirst", "type": "1_n_2", "minMarks": 0, "choices": ["<p>The sequence converges.</p>", "<p>The sequence diverges.</p>", "<p>It doesn't tell us anything about this sequence.</p>"], "showCorrectAnswer": true, "displayType": "dropdownlist", "prompt": "<p><span data-jme-visible=\"aseed=0\">Suppose $\\{a_n\\}$ is a non-decreasing sequence which is bounded above.&nbsp;</span></p>\n<p><span data-jme-visible=\"aseed=1\">Suppose $\\{a_n\\}$ is a non-increasing sequence which is bounded below.&nbsp;</span></p>\n<p><span data-jme-visible=\"aseed=2\">Suppose $\\{a_n\\}$ is a non-decreasing sequence which is bounded.</span></p>\n<p><span data-jme-visible=\"aseed=3\">Suppose $\\{a_n\\}$ is a non-increasing sequence which is bounded.</span></p>\n<p><span data-jme-visible=\"aseed=4\">Suppose $\\{a_n\\}$ is a non-increasing sequence which is bounded above.&nbsp;</span></p>\n<p><span data-jme-visible=\"aseed=5\">Suppose $\\{a_n\\}$ is a non-decreasing sequence which is bounded below.&nbsp;</span></p>\n<p><span data-jme-visible=\"aseed=6\">Suppose $\\{a_n\\}$ is an&nbsp;oscillating sequence which is bounded.&nbsp;</span></p>\n<p></p>\n<p>What does the <strong>monotonic sequence&nbsp;theorem</strong>&nbsp;tell us about the sequence $\\{a_n\\}$?</p>", "distractors": ["", "", ""], "showFeedbackIcon": true, "matrix": ["if(aseed<=3,1,0)", "0", "if(aseed>=4,1,0)"]}, {"scripts": {}, "variableReplacements": [], "marks": 0, "gaps": [{"scripts": {}, "displayColumns": 0, "marks": 0, "shuffleChoices": false, "variableReplacements": [], "maxMarks": 0, "variableReplacementStrategy": "originalfirst", "type": "1_n_2", "minMarks": 0, "choices": ["<p>implies that this sequence is convergent</p>", "<p>does not apply to this sequence</p>"], "showCorrectAnswer": true, "displayType": "dropdownlist", "showFeedbackIcon": true, "matrix": "if(bdict[\"MCT\"]=\"C\",[1,0],[0,1])"}, {"scripts": {}, "displayColumns": 0, "marks": 0, "shuffleChoices": false, "variableReplacements": [], "maxMarks": 0, "variableReplacementStrategy": "originalfirst", "type": "1_n_2", "minMarks": 0, "choices": ["<p>monotonically non-decreasing</p>", "<p>monotonically non-increasing</p>", "<p>not monotonic</p>"], "showCorrectAnswer": true, "displayType": "dropdownlist", "showFeedbackIcon": true, "matrix": "switch(bdict[\"M\"]=\"MND\",[1,0,0],bdict[\"M\"]=\"MNI\",[0,1,0],[0,0,1])"}, {"scripts": {}, "displayColumns": 0, "marks": 0, "shuffleChoices": false, "variableReplacements": [], "maxMarks": 0, "variableReplacementStrategy": "originalfirst", "type": "1_n_2", "minMarks": 0, "choices": ["<p>bounded</p>", "<p>not bounded</p>"], "showCorrectAnswer": true, "displayType": "dropdownlist", "showFeedbackIcon": true, "matrix": "if(bdict[\"B\"]=\"B\",[1,0],[0,1])"}], "showCorrectAnswer": true, "prompt": "<p>Consider the sequence $a_n=${bdict[\"a_n\"]} for $n\\ge1$.</p>\n<p></p>\n<p>The monotonic sequence theorem [[0]]&nbsp;since it is&nbsp;[[1]] and&nbsp;[[2]].</p>", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "type": "gapfill"}], "variables": {"b": {"definition": "coeff[1]", "name": "b", "group": "Ungrouped variables", "description": "", "templateType": "anything"}, "bdictlist": {"definition": "[\n  \n  [\"a_n\":'\\$\\\\displaystyle\\\\simplify{-(n^2+1)/n}\\$',\"B\":\"NB\",\"M\":\"MNI\",\"MCT\":\"NA\"],\n  [\"a_n\":'\\$\\\\displaystyle\\\\simplify{-{a}^n}\\$',\"B\":\"NB\",\"M\":\"MNI\",\"MCT\":\"NA\"],\n  [\"a_n\":'\\$\\\\displaystyle\\\\simplify{-{a}*n+{b}}\\$',\"B\":\"NB\",\"M\":\"MNI\",\"MCT\":\"NA\"],\n  \n  [\"a_n\":'\\$\\\\displaystyle\\\\simplify{(n^2+1)/n}\\$',\"B\":\"NB\",\"M\":\"MND\",\"MCT\":\"NA\"],\n  [\"a_n\":'\\$\\\\displaystyle\\\\simplify{{a}^n}\\$',\"B\":\"NB\",\"M\":\"MND\",\"MCT\":\"NA\"],\n  [\"a_n\":'\\$\\\\displaystyle\\\\simplify{{a}*n+{b}}\\$',\"B\":\"NB\",\"M\":\"MND\",\"MCT\":\"NA\"],\n  \n  [\"a_n\":'\\$\\\\displaystyle\\\\simplify{1/{a}^n}\\$',\"B\":\"B\",\"M\":\"MNI\",\"MCT\":\"C\"],\n  [\"a_n\":'\\$\\\\displaystyle\\\\simplify{1/({a}*n+{b}}\\$',\"B\":\"B\",\"M\":\"MNI\",\"MCT\":\"C\"],\n  [\"a_n\":'\\$\\\\displaystyle\\\\simplify{n/(n^2+1)}\\$',\"B\":\"B\",\"M\":\"MNI\",\"MCT\":\"C\"],\n  \n  [\"a_n\":'\\$\\\\displaystyle\\\\simplify{-1/{a}^n}\\$',\"B\":\"B\",\"M\":\"MND\",\"MCT\":\"C\"],\n  [\"a_n\":'\\$\\\\displaystyle\\\\simplify{-1/({a}*n+{b}}\\$',\"B\":\"B\",\"M\":\"MND\",\"MCT\":\"C\"],\n  [\"a_n\":'\\$\\\\displaystyle\\\\simplify{-n/(n^2+1)}\\$',\"B\":\"B\",\"M\":\"MND\",\"MCT\":\"C\"],\n  \n  \n  [\"a_n\":'\\$\\\\displaystyle\\\\simplify{{a}+(-1)^n/n}\\$',\"B\":\"B\",\"M\":\"NM\",\"MCT\":\"NA\"],\n  [\"a_n\":'\\$\\\\displaystyle\\\\simplify{{a}+sin(n)/n}\\$',\"B\":\"B\",\"M\":\"NM\",\"MCT\":\"NA\"],\n  [\"a_n\":'\\$\\\\displaystyle\\\\simplify{cos({a}n)}\\$',\"B\":\"B\",\"M\":\"NM\",\"MCT\":\"NA\"],\n  \n  [\"a_n\":'\\$\\\\displaystyle\\\\simplify{{a}*n*sin(n)}\\$',\"B\":\"NB\",\"M\":\"NM\",\"MCT\":\"NA\"],\n  [\"a_n\":'\\$\\\\displaystyle\\\\simplify{{a}+(-1)^n*n}\\$',\"B\":\"NB\",\"M\":\"NM\",\"MCT\":\"NA\"],\n  \n  \n  \n  [\"a_n\":'\\$\\\\displaystyle\\\\simplify{{c}-(n^2+1)/n}\\$',\"B\":\"NB\",\"M\":\"MNI\",\"MCT\":\"NA\"],\n  [\"a_n\":'\\$\\\\displaystyle\\\\simplify{{c}-{a}^n}\\$',\"B\":\"NB\",\"M\":\"MNI\",\"MCT\":\"NA\"],\n  [\"a_n\":'\\$\\\\displaystyle\\\\simplify{-{a}*n+{b}}\\$',\"B\":\"NB\",\"M\":\"MNI\",\"MCT\":\"NA\"],\n  \n  [\"a_n\":'\\$\\\\displaystyle\\\\simplify{{c}+(n^2+1)/n}\\$',\"B\":\"NB\",\"M\":\"MND\",\"MCT\":\"NA\"],\n  [\"a_n\":'\\$\\\\displaystyle\\\\simplify{{c}+{a}^n}\\$',\"B\":\"NB\",\"M\":\"MND\",\"MCT\":\"NA\"],\n  [\"a_n\":'\\$\\\\displaystyle\\\\simplify{{a}*n+{b}}\\$',\"B\":\"NB\",\"M\":\"MND\",\"MCT\":\"NA\"],\n  \n  [\"a_n\":'\\$\\\\displaystyle\\\\simplify{{c}+{d}/{a}^n}\\$',\"B\":\"B\",\"M\":\"MNI\",\"MCT\":\"C\"],\n  [\"a_n\":'\\$\\\\displaystyle\\\\simplify{{c}+{d}/({a}*n+{b}}\\$',\"B\":\"B\",\"M\":\"MNI\",\"MCT\":\"C\"],\n  [\"a_n\":'\\$\\\\displaystyle\\\\simplify{{c}+n/(n^2+1)}\\$',\"B\":\"B\",\"M\":\"MNI\",\"MCT\":\"C\"],\n  \n  [\"a_n\":'\\$\\\\displaystyle\\\\simplify{{c}-{d}/{a}^n}\\$',\"B\":\"B\",\"M\":\"MND\",\"MCT\":\"C\"],\n  [\"a_n\":'\\$\\\\displaystyle\\\\simplify{{c}-{d}/({a}*n+{b}}\\$',\"B\":\"B\",\"M\":\"MND\",\"MCT\":\"C\"],\n  [\"a_n\":'\\$\\\\displaystyle\\\\simplify{{c}-n/(n^2+1)}\\$',\"B\":\"B\",\"M\":\"MND\",\"MCT\":\"C\"],\n  \n  \n  [\"a_n\":'\\$\\\\displaystyle\\\\simplify{{a}+(-1)^n/n}\\$',\"B\":\"B\",\"M\":\"NM\",\"MCT\":\"NA\"],\n  [\"a_n\":'\\$\\\\displaystyle\\\\simplify{{a}+sin(n)/n}\\$',\"B\":\"B\",\"M\":\"NM\",\"MCT\":\"NA\"],\n  [\"a_n\":'\\$\\\\displaystyle\\\\simplify{{c}+cos({a}n)}\\$',\"B\":\"B\",\"M\":\"NM\",\"MCT\":\"NA\"],\n  \n  [\"a_n\":'\\$\\\\displaystyle\\\\simplify{{a}*n*sin(n)}\\$',\"B\":\"NB\",\"M\":\"NM\",\"MCT\":\"NA\"],\n  [\"a_n\":'\\$\\\\displaystyle\\\\simplify{{a}+(-1)^n*n}\\$',\"B\":\"NB\",\"M\":\"NM\",\"MCT\":\"NA\"]\n  \n]  ", "name": "bdictlist", "group": "Ungrouped variables", "description": "<p>do more combinations, maybe list both bounds?</p>\n<p></p>\n<p></p>", "templateType": "anything"}, "a": {"definition": "coeff[0]", "name": "a", "group": "Ungrouped variables", "description": "", "templateType": "anything"}, "c": {"definition": "coeff[2]", "name": "c", "group": "Ungrouped variables", "description": "", "templateType": "anything"}, "d": {"definition": "coeff[3]", "name": "d", "group": "Ungrouped variables", "description": "", "templateType": "anything"}, "coeff": {"definition": "shuffle(2..12)[0..4]", "name": "coeff", "group": "Ungrouped variables", "description": "", "templateType": "anything"}, "bdict": {"definition": "random(bdictlist)", "name": "bdict", "group": "Ungrouped variables", "description": "", "templateType": "anything"}, "aseed": {"definition": "random(0..6)", "name": "aseed", "group": "Ungrouped variables", "description": "", "templateType": "anything"}}, "preamble": {"js": "", "css": ""}, "variable_groups": [], "tags": [], "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/"}]}