// Numbas version: exam_results_page_options {"name": "Determine limit of fractional power sequence", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"r": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(100..1000)", "description": "", "name": "r"}, "n": {"templateType": "anything", "group": "Ungrouped variables", "definition": "ceil(s*ln(1/c)/ln((10^t-1)/10^t))", "description": "", "name": "n"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(b1=a,b1+1,b1)", "description": "", "name": "b"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(c1=b or c1=a,max(b,a)+1,c1)", "description": "", "name": "c"}, "s": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "name": "s"}, "b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1000..10000)", "description": "", "name": "b1"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(5000..50000)", "description": "", "name": "a"}, "t": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..4)", "description": "", "name": "t"}, "trn": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s*ln(1/c)/ln((10^t-1)/10^t)", "description": "", "name": "trn"}, "c1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(3..9)", "description": "", "name": "c1"}}, "ungrouped_variables": ["a", "c", "b", "trn", "n", "s", "r", "t", "c1", "b1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Determine limit of fractional power sequence", "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "1", "minValue": "1", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "\n \n \n

Let $\\displaystyle{a_n=\\var{a}^{1/n}}$

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$\\displaystyle{\\lim_{n\\to \\infty} a_n=\\;\\;}$[[0]]

\n \n ", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "1", "minValue": "1", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "\n \n \n

Let $\\displaystyle{b_n=\\var{b}^{\\var{r}/n}}$

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$\\displaystyle{\\lim_{n\\to \\infty} b_n=\\;\\;}$[[0]]

\n \n ", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "1", "minValue": "1", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"type": "numberentry", "correctAnswerFraction": false, "showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "integerAnswer": true, "integerPartialCredit": 0, "minValue": "N", "maxValue": "N", "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "

Consider the sequence $\\displaystyle{c_n=\\left(\\frac{1}{\\var{c}}\\right)^{\\var{s}/n}}$

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1. Let $L=\\displaystyle{\\lim_{n\\to \\infty} c_n}$

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$L=\\;\\;$[[0]]

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2. Find the least positive integer $N$ such that
\\[|c_n-L| \\le 10^{-\\var{t}},\\;\\;\\forall n \\geq N\\]

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Hint: You may assume that  $c_n \\lt L,\\;\\;\\forall n$, hence $|c_n-L|=L-c_n,\\;\\;\\forall n$

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$N=\\;\\;$[[1]]

", "showCorrectAnswer": true, "marks": 0}], "statement": "\n \n \n

We have seen in the notes that $\\displaystyle{2^{1/n} \\longrightarrow 1}$ as $n \\longrightarrow \\infty$.

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Answer the following questions:

\n \n ", "tags": ["checked2015", "finding the limit of a sequence", "limit", "limits", "MAS1601", "sequence", "sequences", "taking the limit", "tested1"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "

5/07/2012:

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Added tags.

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Added explanation in Advice why inequality remains as $\\le$ on taking $\\ln$ of both sides.

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22/07/2012:

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Added description.

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Changed $n \\ge N$ to $n \\geq N$ as otherwise $N-1$ would be be the correct answer not $N$.

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First two parts are trivial.

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27/7/2012:

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Added tags.

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In the Advice section, moved \\Rightarrow to the beginning of the line instead of the end of the previous line.

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Question appears to be working correctly.

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25/12/2012:

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Checked calculation, OK. Added tested1 tag.

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Changed all $\\lt$ etc to $\\le$.  Also edited the description for this.

", "licence": "Creative Commons Attribution 4.0 International", "description": "

Question on $\\displaystyle{\\lim_{n\\to \\infty} a^{1/n}=1}$. Find least integer $N$ s.t.  $\\ \\left |1-\\left(\\frac{1}{c}\\right)^{b/n}\\right| \\le10^{-r},\\;n \\geq N$

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a)

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It is true that for any positive real number $a$ that $\\displaystyle{\\lim_{n\\to \\infty} a^{1/n}=1}$

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Hence $\\displaystyle{\\lim_{n\\to \\infty} \\var{a}^{1/n}=1}$

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b)
We have $\\displaystyle{b_n=\\var{b}^{\\var{r}/n}=\\left(\\var{b}^{\\var{r}}\\right)^{1/n}}$

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Hence by the result quoted in part a) we have $\\displaystyle{\\lim_{n\\to \\infty} b_n=1}$

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c)

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1. Once again by the result quoted in a) and writing $\\displaystyle{\\left(\\frac{1}{\\var{c}} \\right)^{\\var{s}/n}=\\left(\\frac{1}{\\var{c}^{\\var{s}}} \\right)^{1/n}}$

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we have:

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\\[L=\\lim_{n\\to \\infty} c_n = \\lim_{n \\to \\infty}\\left(\\frac{1}{\\var{c}^{\\var{s}}} \\right)^{1/n} = 1\\]

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2. In order to find the least $N$ such that $\\displaystyle{|c_n-1| \\le 10^{-\\var{t}},\\;\\;\\forall n \\ge N}$ we write this using the hint as:

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\\[\\begin{eqnarray*} 1-\\left(\\frac{1}{\\var{c}}\\right)^{\\var{s}/n} &\\le& \\frac{1}{10^{\\var{t}}} \\\\ \\Rightarrow \\left(\\frac{1}{\\var{c}}\\right)^{\\var{s}/n} &\\ge& 1-\\frac{1}{10^{\\var{t}}}=\\frac{\\var{10^t-1}}{\\var{10^t}} \\\\ \\Rightarrow \\var{c}^{\\var{s}/n} &\\le& \\frac{\\var{10^t}}{\\var{10^t-1}} \\end{eqnarray*} \\]

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Taking $\\ln$ of both sides gives:(the inequality sign remains as it is as $\\ln(\\var{c}) \\gt 0 $ and  $\\displaystyle \\ln\\left(\\frac{\\var{10^t}}{\\var{10^t-1}}\\right) \\gt 0$ )

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\\[\\begin{eqnarray*} \\frac{\\var{s}}{n}\\ln(\\var{c}) &\\le& \\ln\\left(\\frac{\\var{10^t}}{\\var{10^t-1}}\\right)\\\\ \\Rightarrow n &\\ge& \\frac{\\var{s}\\ln(\\var{c})}{ \\ln\\left(\\frac{\\var{10^t}}{\\var{10^t-1}}\\right)}=\\var{trN} \\end{eqnarray*} \\]

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On rounding up the least $N$ is $N=\\var{N}$.

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