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

2016-04-19T01:23:23Z

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

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Let

::::::<math>f(x)=\sum_{n=1}^{\infty} nx^n</math>

::a) Find the radius of convergence of the power series.

::b) Determine the interval of convergence of the power series.

::c) Obtain an explicit formula for the function <math style="vertical-align: -5px">f(x)</math>.

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Foundations:  
|-
|Recall:
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::'''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,
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:::If <math style="vertical-align: -1px">L<1,</math> the series is absolutely convergent.
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:::If <math style="vertical-align: -1px">L>1,</math> the series is divergent.
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:::If <math style="vertical-align: -1px">L=1,</math> the test is inconclusive.
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::'''2.''' After you find the radius of convergence, you need to check the endpoints of your interval
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:::for convergence since the Ratio Test is inconclusive when <math style="vertical-align: -1px">L=1.</math>
|}

'''Solution:'''

'''(a)'''

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 1:  
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|To find the radius of convergence, we use the ratio test. We have
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::<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)x^{n+1}}{nx^n}}\bigg|\\
&&\\
& = & \displaystyle{\lim_{n \rightarrow \infty}\bigg|\frac{n+1}{n}x\bigg|}\\
&&\\
& = & \displaystyle{|x|\lim_{n \rightarrow \infty}\frac{n+1}{n}}\\
&&\\
& = & \displaystyle{|x|.}\\
\end{array}</math>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 2:  
|-
|Thus, we have <math style="vertical-align: -5px">|x|<1</math> and the radius of convergence of this series is <math style="vertical-align: -1px">1.</math>
|}

'''(b)'''

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 1:  
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|From part (a), we know the series converges inside the interval <math style="vertical-align: -5px">(-1,1).</math>
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|Now, we need to check the endpoints of the interval for convergence.
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 2:  
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|For <math style="vertical-align: -2px">x=1,</math> the series becomes <math>\sum_{n=1}^{\infty}n,</math> which diverges by the Divergence Test.
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 3:  
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|For <math style="vertical-align: -2px">x=-1,</math> the series becomes <math>\sum_{n=1}^{\infty}(-1)^n n,</math> which diverges by the Divergence Test.
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|Thus, the interval of convergence is <math style="vertical-align: -5px">(-1,1).</math>
|}

'''(c)'''

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 1:  
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|Recall that we have the geometric series formula <math>\frac{1}{1-x}=\sum_{n=0}^{\infty} x^n</math> for <math>|x|<1.</math>
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|Now, we take the derivative of both sides of the last equation to get
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::<math>\frac{1}{(1-x)^2}=\sum_{n=1}^{\infty}nx^{n-1}.</math>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 2:  
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|Now, we multiply the last equation in Step 1 by <math style="vertical-align: 0px">x.</math>
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|So, we have
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::<math>\frac{x}{(1-x)^2}=\sum_{n=1}^{\infty}nx^{n}=f(x).</math>
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|Thus, <math>f(x)=\frac{x}{(1-x)^2}.</math>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Final Answer:  
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|   '''(a)''' <math style="vertical-align: -3px">1</math>
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|   '''(b)''' <math style="vertical-align: -3px">(-1,1)</math>
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|   '''(c)''' <math style="vertical-align: -18px">f(x)=\frac{x}{(1-x)^2}</math>
|}
[[009C_Sample_Final_1|'''Return to Sample Exam''']]

009C Sample Final 1, Problem 5 - Revision history

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

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