Difference between revisions of "009C Sample Midterm 1, Problem 3"

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<span class="exam">Determine whether the following series converges absolutely, conditionally or whether it diverges.
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<span class="exam"> Determine whether the following series converges absolutely,  
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<span class="exam"> conditionally or whether it diverges.
  
 
<span class="exam"> Be sure to justify your answers!
 
<span class="exam"> Be sure to justify your answers!
  
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::::::<math>\sum_{n=1}^\infty \frac{(-1)^n}{n}</math>
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::<math>\sum_{n=1}^\infty \frac{(-1)^n}{n}</math>
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
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!Foundations: &nbsp;
 
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|-
 
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|'''1.''' A series <math>\sum a_n</math> is '''absolutely convergent''' if
 
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|&nbsp; &nbsp; &nbsp; &nbsp; the series <math>\sum |a_n|</math> converges.
 
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|'''2.''' A series <math>\sum a_n</math> is '''conditionally convergent''' if
 
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|&nbsp; &nbsp; &nbsp; &nbsp; the series <math>\sum |a_n|</math> diverges and
 
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|&nbsp; &nbsp; &nbsp; &nbsp; the series <math>\sum a_n</math> converges.
 
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|}
 
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'''Solution:'''
 
  
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
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!Step 1: &nbsp;
 
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|First, we take the absolute value of the terms in the original series.
 
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|Let <math>a_n=\frac{(-1)^n}{n}.</math>
 
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|Therefore,
 
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|&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
 
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\displaystyle{\sum_{n=1}^\infty |a_n|} & = & \displaystyle{\sum_{n=1}^\infty \bigg|\frac{(-1)^n}{n}\bigg|}\\
 
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&&\\
 
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& = & \displaystyle{\sum_{n=1}^\infty \frac{1}{n}.}
 
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\end{array}</math>
 
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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<hr>
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!Step 2: &nbsp;
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[[009C Sample Midterm 1, Problem 3 Solution|'''<u>Solution</u>''']]
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|This series is the harmonic series (or <math>p</math>-series with <math>p=1</math>).
 
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|So, it diverges. Hence the series
 
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|&nbsp; &nbsp; &nbsp; &nbsp; <math>\sum_{n=1}^\infty \frac{(-1)^n}{n}</math>
 
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|is not absolutely convergent.
 
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
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!Step 3: &nbsp;
 
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|Now, we need to look back at the original series to see
 
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|if it is conditionally converges.
 
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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[[009C Sample Midterm 1, Problem 3 Detailed Solution|'''<u>Detailed Solution</u>''']]
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!Step 4: &nbsp;
 
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
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!Final Answer: &nbsp;
 
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[[009C_Sample_Midterm_1|'''<u>Return to Sample Exam</u>''']]
 
[[009C_Sample_Midterm_1|'''<u>Return to Sample Exam</u>''']]

Latest revision as of 22:54, 11 November 2017

Determine whether the following series converges absolutely,

conditionally or whether it diverges.

Be sure to justify your answers!


Solution


Detailed Solution


Return to Sample Exam

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