009C Sample Midterm 3, Problem 4 - Revision history

MathAdmin at 16:46, 28 November 2017

2017-11-28T16:46:37Z

MathAdmin at 22:21, 26 April 2015

2015-04-26T22:21:35Z

MathAdmin: Created page with "Test the series for convergence or divergence. ::(a) (6 points) ${\displaystyle \sum_{n=1}^{\infty}}\,(-1..."$

2015-04-26T22:09:41Z

Created page with "Test the series for convergence or divergence. ::(a) (6 points) <math>{\displaystyle \sum_{n=1}^{\infty}}\,(-1..."

New page

Test the series for convergence or divergence.

::(a) (6 points)      <math>{\displaystyle \sum_{n=1}^{\infty}}\,(-1)^{n}\sin\frac{\pi}{n}.</math>

::(b) (6 points)      <math>{\displaystyle \sum_{n=1}^{\infty}}\,(-1)^{n}\cos\frac{\pi}{n}.</math>

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
! Foundations:  
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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 ''not'' 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.
|}

 '''Solution:'''

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!(a):  
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|Here, we have
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::<math>placehold</math>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!(b):  
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|

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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Final Answer:  
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|}
[[009C_Sample_Midterm_3|'''Return to Sample Exam''']]

@@ Line 1: / Line 1: @@
 <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>
+<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>
+<span class="exam">(b) &nbsp;<math>{\displaystyle \sum_{n=1}^{\infty}}\,(-1)^{n}\cos\frac{\pi}{n}</math>
-&nbsp;'''Solution:'''
+[[009C Sample Midterm 3, Problem 4 Detailed Solution|'''<u>Detailed Solution</u>''']]
 [[009C_Sample_Midterm_3|'''<u>Return to Sample Exam</u>''']]

@@ Line 36: / Line 36: @@
 |-
+|
-::<math>placehold</math>
+::<math>\begin{array}{rcl}
 |}
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 !(b): &nbsp;
 |-
+|
+::<math>\begin{array}{rcl}
 |}
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 !Final Answer: &nbsp;
 |-
+|The series for (a) is convergent, while the series (b) is divergent.
 |}
 [[009C_Sample_Midterm_3|'''<u>Return to Sample Exam</u>''']]

009C Sample Midterm 3, Problem 4 - Revision history

MathAdmin at 16:46, 28 November 2017

MathAdmin at 22:21, 26 April 2015

MathAdmin: Created page with "Test the series for convergence or divergence. ::(a) (6 points) {\displaystyle \sum_{n=1}^{\infty}}\,(-1..."

MathAdmin: Created page with "Test the series for convergence or divergence. ::(a) (6 points) ${\displaystyle \sum_{n=1}^{\infty}}\,(-1..."$