Difference between revisions of "009C Sample Final 2, Problem 4"

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(Replaced content with "<span class="exam">(a) Find the radius of convergence for the power series ::<math>\sum_{n=1}^{\infty} (-1)^n \frac{x^n}{n}.</math> <span class="exam">(b) Find the inter...")
 
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<span class="exam">(a) Find the radius of convergence for the power series
 
<span class="exam">(a) Find the radius of convergence for the power series
  
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::<math>\sum_{n=1}^{+\infty} (-1)^n \frac{x^n}{n}.</math>
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::<math>\sum_{n=1}^{\infty} (-1)^n \frac{x^n}{n}.</math>
  
 
<span class="exam">(b) Find the interval of convergence of the above series.
 
<span class="exam">(b) Find the interval of convergence of the above series.
  
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!Foundations: &nbsp;
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[[009C Sample Final 2, Problem 4 Solution|'''<u>Solution</u>''']]
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|'''Ratio Test'''
 
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|&nbsp; &nbsp; &nbsp; &nbsp; Let &nbsp;<math style="vertical-align: -7px">\sum a_n</math>&nbsp; be a series and &nbsp;<math>L=\lim_{n\rightarrow \infty}\bigg|\frac{a_{n+1}}{a_n}\bigg|.</math>
 
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|&nbsp; &nbsp; &nbsp; &nbsp; Then,
 
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&nbsp; &nbsp; &nbsp; &nbsp; If &nbsp;<math style="vertical-align: -4px">L<1,</math>&nbsp; the series is absolutely convergent.
 
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&nbsp; &nbsp; &nbsp; &nbsp; If &nbsp;<math style="vertical-align: -4px">L>1,</math>&nbsp; the series is divergent.
 
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&nbsp; &nbsp; &nbsp; &nbsp; If &nbsp;<math style="vertical-align: -4px">L=1,</math>&nbsp; the test is inconclusive.
 
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'''Solution:'''
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[[009C Sample Final 2, Problem 4 Detailed Solution|'''<u>Detailed Solution</u>''']]
  
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'''(a)'''
 
  
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!Step 1: &nbsp;
 
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!Step 2: &nbsp;
 
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'''(b)'''
 
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!Step 1: &nbsp;
 
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!Final Answer: &nbsp;
 
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|&nbsp;&nbsp; '''(a)'''
 
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|&nbsp;&nbsp; '''(b)'''
 
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[[009C_Sample_Final_2|'''<u>Return to Sample Exam</u>''']]
 
[[009C_Sample_Final_2|'''<u>Return to Sample Exam</u>''']]

Latest revision as of 18:27, 2 December 2017

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

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

Solution


Detailed Solution


Return to Sample Exam

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