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| | <span class="exam">Test the series for convergence or divergence. | | <span class="exam">Test the series for convergence or divergence. |
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| − | ::<span class="exam">(a) (6 points) <math>{\displaystyle \sum_{n=1}^{\infty}}\,(-1)^{n}\sin\frac{\pi}{n}.</math>
| + | <span class="exam">(a) <math>{\displaystyle \sum_{n=1}^{\infty}}\,(-1)^{n}\sin\frac{\pi}{n}</math> |
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| − | ::<span class="exam">(b) (6 points) <math>{\displaystyle \sum_{n=1}^{\infty}}\,(-1)^{n}\cos\frac{\pi}{n}.</math>
| + | <span class="exam">(b) <math>{\displaystyle \sum_{n=1}^{\infty}}\,(-1)^{n}\cos\frac{\pi}{n}</math> |
| | + | <hr> |
| | + | [[009C Sample Midterm 3, Problem 4 Solution|'''<u>Solution</u>''']] |
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | ! Foundations:
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| − | |-
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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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| − | |-
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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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| − | |}
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| − | '''Solution:'''
| + | [[009C Sample Midterm 3, Problem 4 Detailed Solution|'''<u>Detailed Solution</u>''']] |
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !(a):
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| − | |-
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| − | |Here, we have
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| − | ::<math>placehold</math>
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| − | |}
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !(b):
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| − | |}
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Final Answer:
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| − | |}
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| | [[009C_Sample_Midterm_3|'''<u>Return to Sample Exam</u>''']] | | [[009C_Sample_Midterm_3|'''<u>Return to Sample Exam</u>''']] |
Test the series for convergence or divergence.
(a) 
(b) 
Solution
Detailed Solution
Return to Sample Exam