Difference between revisions of "009C Sample Final 2, Problem 3"

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<span class="exam">Determine if the following series converges or diverges. Please give your reason(s).
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<span class="exam"> Determine if the following series converges or diverges. Please give your reason(s).
  
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<span class="exam">(a) &nbsp;<math>\sum_{n=0}^{+\infty} \frac{n!}{(2n)!}</math>
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<span class="exam">(a) &nbsp;<math>\sum_{n=1}^{\infty} \frac{n!}{(2n)!}</math>  
  
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<span class="exam">(b) &nbsp;<math>\sum_{n=0}^{+\infty} (-1)^n \frac{1}{n+1}</math>
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<span class="exam">(b) &nbsp;<math>\sum_{n=1}^{\infty} (-1)^n\frac{1}{n+1}</math>
  
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!Foundations: &nbsp;
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[[009C Sample Final 2, Problem 3 Solution|'''<u>Solution</u>''']]
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'''Solution:'''
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[[009C Sample Final 2, Problem 3 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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|For
 
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|&nbsp; &nbsp; &nbsp; &nbsp; <math>\sum_{n=1}^\infty (-1)^n\frac{1}{n+1},</math>
 
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|we notice that this series is alternating.
 
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|Let &nbsp;<math style="vertical-align: -16px"> b_n=\frac{1}{n+1}.</math>
 
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|The sequence &nbsp;<math style="vertical-align: -5px">\{b_n\}</math>&nbsp; is decreasing since
 
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|&nbsp; &nbsp; &nbsp; &nbsp; <math>\frac{1}{n+2}<\frac{1}{n+1}</math>
 
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|for all &nbsp;<math style="vertical-align: -3px">n\ge 0.</math>
 
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!Step 2: &nbsp;
 
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|Also,
 
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|&nbsp; &nbsp; &nbsp; &nbsp; <math>\lim_{n\rightarrow \infty}b_n=\lim_{n\rightarrow \infty}\frac{1}{n+1}=0.</math>
 
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|Therefore, the series &nbsp;<math>\sum_{n=1}^\infty (-1)^n\frac{1}{n+1}</math> &nbsp; converges
 
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|by the Alternating Series Test.
 
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!Final Answer: &nbsp;
 
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|&nbsp;&nbsp; '''(a)'''
 
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|&nbsp;&nbsp; '''(b)''' &nbsp;&nbsp; converges
 
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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 18:24, 2 December 2017

Determine if the following series converges or diverges. Please give your reason(s).

(a)  

(b)  

Solution


Detailed Solution


Return to Sample Exam

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