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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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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
| + | <hr> |
| − | !Foundations:
| + | [[009C Sample Final 2, Problem 7 Solution|'''<u>Solution</u>''']] |
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| − | |'''1.''' The Taylor polynomial of <math style="vertical-align: -5px">f(x)</math> at <math style="vertical-align: -1px">a</math> is
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| − | <math>\sum_{n=0}^{\infty}c_n(x-a)^n</math> where <math style="vertical-align: -14px">c_n=\frac{f^{(n)}(a)}{n!}.</math>
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| − | | '''2.''' '''Ratio Test'''
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| − | | 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>
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| − | | Then,
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| − | If <math style="vertical-align: -4px">L<1,</math> the series is absolutely convergent.
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| − | If <math style="vertical-align: -4px">L>1,</math> the series is divergent.
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| − | If <math style="vertical-align: -4px">L=1,</math> the test is inconclusive.
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| − | |}
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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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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Step 1:
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| − | |We begin by finding the coefficients of the Maclaurin series for <math>f(x)=\frac{1}{(1-\frac{1}{2}x)^2}.</math>
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| − | |We make a table to find the coefficients of the Maclaurin series.
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| − | <table border="1" cellspacing="0" cellpadding="11" align = "center">
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| − | <tr>
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| − | <td align = "center"><math> n</math></td>
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| − | <td align = "center"><math> f^{(n)}(x) </math></td>
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| − | <td align = "center"><math> f^{(n)}(0) </math></td>
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| − | <td align = "center"><math> \frac{f^{(n)}(0)}{n!} </math></td>
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| − | </tr>
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| − | <tr>
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| − | <td align = "center"><math>0</math></td>
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| − | <td align = "center"><math> \frac{1}{(1-\frac{1}{2}x)^2} </math></td>
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| − | <td align = "center"><math> 1</math></td>
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| − | <td align = "center"><math> 1</math></td>
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| − | </tr>
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| − | <tr>
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| − | <td align = "center"><math>1</math></td>
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| − | <td align = "center"><math> \frac{1}{(1-\frac{1}{2}x)^3}</math></td>
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| − | <td align = "center"><math> 1 </math></td>
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| − | <td align = "center"><math> 1</math></td>
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| − | </tr>
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| − | <tr>
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| − | <td align = "center"><math>2</math></td>
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| − | <td align = "center"><math> \frac{\frac{3}{2}}{(1-\frac{1}{2}x)^4} </math></td>
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| − | <td align = "center"><math> \frac{3}{2} </math></td>
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| − | <td align = "center"><math> \frac{3}{4} </math></td>
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| − | </tr>
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| − | </table>
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| − | |}
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Step 2:
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| − | |So, the first three terms of the Binomial Series is
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| − | | <math>1+x+\frac{3}{4}x^2.</math>
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| − | |}
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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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| − | |}
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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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| − | |}
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Final Answer:
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| − | | '''(a)''' <math>1+x+\frac{3}{4}x^2</math>
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| − | | '''(b)''' The radius of convergence is <math style="vertical-align: 0px">R=2.</math>
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| − | |}
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| | [[009C_Sample_Final_2|'''<u>Return to Sample Exam</u>''']] | | [[009C_Sample_Final_2|'''<u>Return to Sample Exam</u>''']] |
