Difference between revisions of "009C Sample Final 3, Problem 6"

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<span class="exam"> Consider the power series  
 
<span class="exam"> Consider the power series  
  
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::::<math>\sum_{n=0}^\infty (-1)^n \frac{x^{n+1}}{n+1}</math>
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::<math>\sum_{n=0}^\infty (-1)^n \frac{x^{n+1}}{n+1}</math>
  
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::<span class="exam">a) Find the radius of convergence of the above power series.
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<span class="exam">(a) Find the radius of convergence of the above power series.
  
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::<span class="exam">b) Find the interval of convergence of the above power series.
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<span class="exam">(b) Find the interval of convergence of the above power series.
  
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::<span class="exam">c) Find the closed formula for the function <math>f(x)</math> to which the power series converges.
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<span class="exam">(c) Find the closed formula for the function &nbsp;<math>f(x)</math>&nbsp; to which the power series converges.
  
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::<span class="exam">d) Does the series
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<span class="exam">(d) Does the series
  
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::::<math>\sum_{n=0}^\infty \frac{1}{(n+1)3^{n+1}}</math>
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::<math>\sum_{n=0}^\infty \frac{1}{(n+1)3^{n+1}}</math>
  
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::<span class="exam">converge? If so, find its sum.
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<span class="exam">converge? If so, find its sum.
  
 
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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'''Solution:'''
 
'''Solution:'''
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'''(c)'''
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!Step 1: &nbsp;
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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!Step 2: &nbsp;
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'''(d)'''
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!Step 1: &nbsp;
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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!Step 2: &nbsp;
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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|&nbsp;&nbsp; '''(a)'''  
 
|&nbsp;&nbsp; '''(a)'''  
 
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|&nbsp;&nbsp; '''(b)'''  
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|&nbsp;&nbsp; '''(b)'''
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|&nbsp;&nbsp; '''(c)'''
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|&nbsp;&nbsp; '''(d)'''  
 
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[[009C_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]
 
[[009C_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]

Revision as of 17:58, 4 March 2017

Consider the power series

(a) Find the radius of convergence of the above power series.

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

(c) Find the closed formula for the function    to which the power series converges.

(d) Does the series

converge? If so, find its sum.

Foundations:  


Solution:

(a)

Step 1:  
Step 2:  

(b)

Step 1:  
Step 2:  

(c)

Step 1:  
Step 2:  

(d)

Step 1:  
Step 2:  


Final Answer:  
   (a)
   (b)
   (c)
   (d)

Return to Sample Exam

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