Difference between revisions of "009C Sample Midterm 3, Problem 4"

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(Created page with "<span class="exam">Test the series for convergence or divergence. ::<span class="exam">(a) (6 points)      <math>{\displaystyle \sum_{n=1}^{\infty}}\,(-1...")
 
 
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<span class="exam">Test the series for convergence or divergence.  
 
<span class="exam">Test the series for convergence or divergence.  
  
::<span class="exam">(a) (6 points) &nbsp;&nbsp;&nbsp;&nbsp; <math>{\displaystyle \sum_{n=1}^{\infty}}\,(-1)^{n}\sin\frac{\pi}{n}.</math>
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<span class="exam">(a) &nbsp;<math>{\displaystyle \sum_{n=1}^{\infty}}\,(-1)^{n}\sin\frac{\pi}{n}</math>
  
::<span class="exam">(b) (6 points) &nbsp;&nbsp;&nbsp;&nbsp; <math>{\displaystyle \sum_{n=1}^{\infty}}\,(-1)^{n}\cos\frac{\pi}{n}.</math>
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<span class="exam">(b) &nbsp;<math>{\displaystyle \sum_{n=1}^{\infty}}\,(-1)^{n}\cos\frac{\pi}{n}</math>
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<hr>
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[[009C Sample Midterm 3, Problem 4 Solution|'''<u>Solution</u>''']]
  
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
! Foundations: &nbsp;
 
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|For <math>n\geq2</math>, both sine and cosine of <math>\frac{\pi}{n}</math> are strictly nonnegative.  Thus, these series are alternating, and we can apply the
 
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|'''Alternating Series Test:''' If a series <math>\sum_{k=1}^{\infty} a_{k}</math> is
 
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:*Alternating in sign, and
 
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:*<math>\lim_{k\rightarrow 0}|a_{k}|=0,</math>
 
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|then the series is convergent.
 
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|Note that if the series does ''<u>not</u>'' converge to zero, we must claim it diverges by the
 
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'''Divergence Test:''' If <math style="vertical-align: -65%">{\displaystyle \lim_{k\rightarrow\infty}a_{k}\neq0,}</math> then the series/sum  <math style="vertical-align: -98%">\sum_{k=0}^{\infty}a_{k}</math> diverges.
 
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|In the case of an alternating series, such as the two listed for this problem, we can choose to show it does not converge to zero absolutely.
 
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&nbsp;'''Solution:'''
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[[009C Sample Midterm 3, Problem 4 Detailed Solution|'''<u>Detailed Solution</u>''']]
  
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!(a): &nbsp;
 
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|Here, we have
 
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::<math>placehold</math>
 
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!(b): &nbsp;
 
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Final Answer: &nbsp;
 
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[[009C_Sample_Midterm_3|'''<u>Return to Sample Exam</u>''']]
 
[[009C_Sample_Midterm_3|'''<u>Return to Sample Exam</u>''']]

Latest revision as of 09:46, 28 November 2017

Test the series for convergence or divergence.

(a)  

(b)  


Solution


Detailed Solution


Return to Sample Exam