MathAdmin: Created page with "(a) Find the radius of convergence for the power series :: $\sum_{n=1}^{\infty} (-1)^n \frac{x^n}{n}.$ (b) Find the interval..."

2017-03-12T16:52:33Z

Created page with "(a) Find the radius of convergence for the power series ::<math>\sum_{n=1}^{\infty} (-1)^n \frac{x^n}{n}.</math> (b) Find the interval..."

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(a) Find the radius of convergence for the power series

::<math>\sum_{n=1}^{\infty} (-1)^n \frac{x^n}{n}.</math>

(b) Find the interval of convergence of the above series.

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Foundations:  
|-
|'''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.
|}

'''Solution:'''

'''(a)'''

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 1:  
|-
|We use the Ratio Test to determine the radius of convergence.
|-
|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{(-1)^{n+1}(x)^{n+1}}{(n+1)}\cdot\frac{n}{(-1)^n(x)^n}\bigg|}\\
&&\\
& = & \displaystyle{\lim_{n\rightarrow \infty} \bigg|(-1)(x)\frac{n}{n+1}\bigg|}\\
&&\\
& = & \displaystyle{\lim_{n\rightarrow \infty} |x|\frac{n}{n+1}}\\
&&\\
& = & \displaystyle{|x|\lim_{n\rightarrow \infty} \frac{n}{n+1}}\\
&&\\
& = & \displaystyle{|x|.}
\end{array}</math>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 2:  
|-
|The Ratio Test tells us this series is absolutely convergent if  <math style="vertical-align: -5px">|x|<1.</math>
|-
|Hence, the Radius of Convergence of this series is  <math style="vertical-align: -1px">R=1.</math>
|}

'''(b)'''

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 1:  
|-
|First, note that  <math style="vertical-align: -5px">|x|<1</math>  corresponds to the interval  <math style="vertical-align: -4px">(-1,1).</math>
|-
|To obtain the interval of convergence, we need to test the endpoints of this interval
|-
|for convergence since the Ratio Test is inconclusive when  <math style="vertical-align: -1px">R=1.</math>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 2:  
|-
|First, let  <math style="vertical-align: -1px">x=1.</math>
|-
|Then, the series becomes  <math>\sum_{n=0}^\infty (-1)^n \frac{1}{n}.</math>
|-
|This is an alternating series.
|-
|Let  <math style="vertical-align: -15px">b_n=\frac{1}{n}.</math>.
|-
|The sequence  <math>\{b_n\}</math>  is decreasing since
|-
|        <math>\frac{1}{n+1}<\frac{1}{n}</math>
|-
|for all  <math style="vertical-align: -3px">n\ge 1.</math>
|-
|Also,
|-
|        <math>\lim_{n\rightarrow \infty} b_n=\lim_{n\rightarrow \infty} \frac{1}{n}=0.</math>
|-
|Therefore, this series converges by the Alternating Series Test
|-
|and we include  <math style="vertical-align: -1px">x=1</math>  in our interval.
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 3:  
|-
|Now, let  <math style="vertical-align: -1px">x=-1.</math>
|-
|Then, the series becomes  <math>\sum_{n=0}^\infty \frac{1}{n}.</math>
|-
|This is a  <math>p</math>-series with  <math>p=1.</math>  Hence, the series diverges.
|-
|Therefore, we do not include  <math style="vertical-align: -1px">x=-1</math>  in our interval.
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 4:  
|-
|The interval of convergence is  <math style="vertical-align: -4px">(-1,1].</math>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Final Answer:  
|-
|    '''(a)'''     The radius of convergence is  <math style="vertical-align: -1px">R=1.</math>
|-
|    '''(b)'''     <math>(-1,1]</math>
|}
[[009C_Sample_Final_2|'''Return to Sample Exam''']]

009C Sample Final 2, Problem 4 - Revision history

MathAdmin: Created page with "(a) Find the radius of convergence for the power series ::\sum_{n=1}^{\infty} (-1)^n \frac{x^n}{n}. (b) Find the interval..."

MathAdmin: Created page with "(a) Find the radius of convergence for the power series :: $\sum_{n=1}^{\infty} (-1)^n \frac{x^n}{n}.$ (b) Find the interval..."