MathAdmin: Created page with "Determine if the following series converges or diverges. Please give your reason(s). (a) $\sum_{n=0}^{\infty} \frac{n!}{(2n)!..."$

2017-03-12T16:51:18Z

Created page with "Determine if the following series converges or diverges. Please give your reason(s). (a) <math>\sum_{n=0}^{\infty} \frac{n!}{(2n)!..."

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

(a)  <math>\sum_{n=0}^{\infty} \frac{n!}{(2n)!}</math>

(b)  <math>\sum_{n=0}^{\infty} (-1)^n \frac{1}{n+1}</math>

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Foundations:  
|-
|'''1.''' '''Ratio Test'''
|-
|        Let  <math style="vertical-align: -7px">\sum a_n</math>  be a series and  <math>L=\lim_{n\rightarrow \infty}\bigg|\frac{a_{n+1}}{a_n}\bigg|.</math>
|-
|        Then,
|-
|
        If  <math style="vertical-align: -4px">L<1,</math>  the series is absolutely convergent.
|-
|
        If  <math style="vertical-align: -4px">L>1,</math>  the series is divergent.
|-
|
        If  <math style="vertical-align: -4px">L=1,</math>  the test is inconclusive.
|-
|'''2.''' If a series absolutely converges, then it also converges.
|-
|'''3.''' '''Alternating Series Test'''
|-
|        Let  <math>\{a_n\}</math>  be a positive, decreasing sequence where  <math style="vertical-align: -11px">\lim_{n\rightarrow \infty} a_n=0.</math>
|-
|        Then,  <math>\sum_{n=1}^\infty (-1)^na_n</math>  and  <math>\sum_{n=1}^\infty (-1)^{n+1}a_n</math>
|-
|        converge.
|}

'''Solution:'''

'''(a)'''

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 1:  
|-
|We begin by using the Ratio Test.
|-
|We have
|-
|
        <math>\begin{array}{rcl}
\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))!} \cdot\frac{(2n)!}{n!}\bigg|}\\
&&\\
& = & \displaystyle{\lim_{n\rightarrow \infty} \bigg| \frac{(n+1)n!}{(2n+2)(2n+1)(2n)!} \cdot \frac{(2n)!}{n!}\bigg|}\\
&&\\
& = & \displaystyle{\lim_{n\rightarrow \infty} \frac{n+1}{(2n+2)(2n+1)}}\\
&&\\
& = & \displaystyle{0.}
\end{array}</math>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 2:  
|-
|Since
|-
|        <math>\lim_{n\rightarrow \infty} \bigg|\frac{a_{n+1}}{a_n}\bigg|=0<1,</math>
|-
|the series is absolutely convergent by the Ratio Test.
|-
|Therefore, the series converges.
|}

'''(b)'''

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 1:  
|-
|For
|-
|        <math>\sum_{n=1}^\infty (-1)^n\frac{1}{n+1},</math>
|-
|we notice that this series is alternating.
|-
|Let  <math style="vertical-align: -16px"> b_n=\frac{1}{n+1}.</math>
|-
|The sequence  <math style="vertical-align: -5px">\{b_n\}</math>  is decreasing since
|-
|        <math>\frac{1}{n+2}<\frac{1}{n+1}</math>
|-
|for all  <math style="vertical-align: -3px">n\ge 0.</math>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 2:  
|-
|Also,
|-
|        <math>\lim_{n\rightarrow \infty}b_n=\lim_{n\rightarrow \infty}\frac{1}{n+1}=0.</math>
|-
|Therefore, the series  <math>\sum_{n=1}^\infty (-1)^n\frac{1}{n+1}</math>   converges
|-
|by the Alternating Series Test.
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Final Answer:  
|-
|   '''(a)'''    converges
|-
|   '''(b)'''    converges
|}
[[009C_Sample_Final_2|'''Return to Sample Exam''']]

009C Sample Final 2, Problem 3 - Revision history

MathAdmin: Created page with "Determine if the following series converges or diverges. Please give your reason(s). (a) \sum_{n=0}^{\infty} \frac{n!}{(2n)!..."

MathAdmin: Created page with "Determine if the following series converges or diverges. Please give your reason(s). (a) $\sum_{n=0}^{\infty} \frac{n!}{(2n)!..."$