Difference between revisions of "009C Sample Final 3, Problem 4"

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<span class="exam">(b) &nbsp;<math>\sum_{n=1}^{\infty} (-1)^n\frac{1}{n+1}</math>
 
<span class="exam">(b) &nbsp;<math>\sum_{n=1}^{\infty} (-1)^n\frac{1}{n+1}</math>
  
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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<hr>
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!Foundations: &nbsp;
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[[009C Sample Final 3, Problem 4 Solution|'''<u>Solution</u>''']]
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|-
 
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|'''1.''' '''Ratio Test'''
 
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|&nbsp; &nbsp; &nbsp; &nbsp; Let &nbsp;<math style="vertical-align: -7px">\sum a_n</math>&nbsp; be a series and &nbsp;<math>L=\lim_{n\rightarrow \infty}\bigg|\frac{a_{n+1}}{a_n}\bigg|.</math>  
 
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|&nbsp; &nbsp; &nbsp; &nbsp; Then,
 
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|
 
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&nbsp; &nbsp; &nbsp; &nbsp; If &nbsp;<math style="vertical-align: -4px">L<1,</math>&nbsp; the series is absolutely convergent.
 
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&nbsp; &nbsp; &nbsp; &nbsp; If &nbsp;<math style="vertical-align: -4px">L>1,</math>&nbsp; the series is divergent.
 
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&nbsp; &nbsp; &nbsp; &nbsp; If &nbsp;<math style="vertical-align: -4px">L=1,</math>&nbsp; the test is inconclusive.
 
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|'''2.''' If a series absolutely converges, then it also converges.
 
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|'''3.''' '''Alternating Series Test'''
 
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|&nbsp; &nbsp; &nbsp; &nbsp; Let &nbsp;<math>\{a_n\}</math>&nbsp; be a positive, decreasing sequence where &nbsp;<math style="vertical-align: -11px">\lim_{n\rightarrow \infty} a_n=0.</math>
 
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|&nbsp; &nbsp; &nbsp; &nbsp; Then, &nbsp;<math>\sum_{n=1}^\infty (-1)^na_n</math>&nbsp; and &nbsp;<math>\sum_{n=1}^\infty (-1)^{n+1}a_n</math>
 
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|&nbsp; &nbsp; &nbsp; &nbsp; converge.
 
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'''Solution:'''
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[[009C Sample Final 3, Problem 4 Detailed Solution|'''<u>Detailed Solution</u>''']]
  
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'''(a)'''
 
  
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
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!Step 1: &nbsp;
 
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|-
 
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|We begin by using the Ratio Test.
 
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|We have
 
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&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
 
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\displaystyle{\lim_{n\rightarrow \infty}\bigg|\frac{a_{n+1}}{a_n}\bigg|} & = & \displaystyle{\lim_{n\rightarrow \infty} \bigg| \frac{(n+1)!}{(2(n+1))!} \frac{(2n)!}{n!}\bigg|}\\
 
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&&\\
 
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& = & \displaystyle{\lim_{n\rightarrow \infty} \bigg| \frac{(n+1)n!}{(2n+2)(2n+1)(2n)!} \frac{(2n)!}{n!}\bigg|}\\
 
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&&\\
 
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& = & \displaystyle{\lim_{n\rightarrow \infty} \frac{n+1}{(2n+2)(2n+1)}}\\
 
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&&\\
 
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& = & \displaystyle{0.}
 
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\end{array}</math>
 
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|}
 
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
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!Step 2: &nbsp;
 
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|Since
 
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|&nbsp; &nbsp; &nbsp; &nbsp; <math>\lim_{n\rightarrow \infty} \bigg|\frac{a_{n+1}}{a_n}\bigg|=0<1,</math>
 
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|the series is absolutely convergent by the Ratio Test.
 
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|Therefore, the series converges.
 
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'''(b)'''
 
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!Step 1: &nbsp;
 
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|For
 
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|&nbsp; &nbsp; &nbsp; &nbsp; <math>\sum_{n=1}^\infty (-1)^n\frac{1}{n+1},</math>
 
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|we notice that this series is alternating.
 
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|Let &nbsp;<math style="vertical-align: -16px"> b_n=\frac{1}{n+1}.</math>
 
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|First, we have
 
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|&nbsp; &nbsp; &nbsp; &nbsp;<math>\frac{1}{n+1}\ge 0</math>
 
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|for all &nbsp;<math style="vertical-align: -3px">n\ge 1.</math>
 
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|The sequence &nbsp;<math style="vertical-align: -5px">\{b_n\}</math>&nbsp; is decreasing since
 
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|&nbsp; &nbsp; &nbsp; &nbsp; <math>\frac{1}{n+2}<\frac{1}{n+1}</math>
 
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|for all &nbsp;<math style="vertical-align: -3px">n\ge 1.</math>
 
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|}
 
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
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!Step 2: &nbsp;
 
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|Also,
 
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|&nbsp; &nbsp; &nbsp; &nbsp; <math>\lim_{n\rightarrow \infty}b_n=\lim_{n\rightarrow \infty}\frac{1}{n+1}=0.</math>
 
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|Therefore, the series &nbsp;<math>\sum_{n=1}^\infty (-1)^n\frac{1}{n+1}</math> &nbsp; converges
 
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|by the Alternating Series Test.
 
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
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!Final Answer: &nbsp;
 
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|&nbsp;&nbsp; '''(a)''' &nbsp;&nbsp; converges
 
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|&nbsp;&nbsp; '''(b)''' &nbsp;&nbsp; converges
 
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[[009C_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]
 
[[009C_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]

Latest revision as of 18:45, 2 December 2017

Determine if the following series converges or diverges. Please give your reason(s).

(a)  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=1}^{\infty} \frac{n!}{(2n)!}}

(b)  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=1}^{\infty} (-1)^n\frac{1}{n+1}}

Solution


Detailed Solution


Return to Sample Exam

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