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| | <span class="exam">(b) Find its radius of convergence. | | <span class="exam">(b) Find its radius of convergence. |
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| − | !Foundations:
| + | [[009C Sample Final 2, Problem 7 Solution|'''<u>Solution</u>''']] |
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| − | '''Solution:''' | + | [[009C Sample Final 2, Problem 7 Detailed Solution|'''<u>Detailed Solution</u>''']] |
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| − | '''(a)'''
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| − | !Step 1:
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Step 2:
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| − | '''(b)'''
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Step 1:
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| − | |The Maclaurin series of <math>\frac{1}{(1-x)^2}</math> is
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| − | | <math>\sum_{n=0}^\infty (n+1)x^n.</math>
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| − | |So, the Maclaurin series of <math>\frac{1}{(1-\frac{1}{2}x)^2}</math> is
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| − | | <math>\sum_{n=0}^\infty (n+1)\bigg(\frac{1}{2}x\bigg)^n=\sum_{n=0}^\infty \frac{(n+1)x^n}{2^n}.</math>
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Step 2:
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| − | |Now, we use the Ratio Test to determine the radius of convergence of this power series.
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| − | |We have
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| − | | <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+2)x^{n+1}}{2^{n+1}} \frac{2^n}{(n+1)x^n}\bigg|}\\
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| − | &&\\
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| − | & = & \displaystyle{\lim_{n\rightarrow \infty} \frac{|x|}{2} \frac{n+2}{n+1}}\\
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| − | &&\\
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| − | & = & \displaystyle{\frac{|x|}{2}\lim_{n\rightarrow \infty}\frac{n+2}{n+1}}\\
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| − | &&\\
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| − | & = & \displaystyle{\frac{|x|}{2}.}
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| − | \end{array}</math>
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| − | |Now, the Ratio Test says this series converges if <math style="vertical-align: -14px">\frac{|x|}{2}<1.</math> So, <math style="vertical-align: -6px">|x|<2.</math>
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| − | |Hence, the radius of convergence is <math style="vertical-align: 0px">R=2.</math>
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Final Answer:
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| − | | '''(a)'''
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| − | | '''(b)''' The radius of convergence is <math style="vertical-align: 0px">R=2.</math>
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| | [[009C_Sample_Final_2|'''<u>Return to Sample Exam</u>''']] | | [[009C_Sample_Final_2|'''<u>Return to Sample Exam</u>''']] |
