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| | <span class="exam">(b) Test if the series converges conditionally. Give reasons for your answer. | | <span class="exam">(b) Test if the series converges conditionally. Give reasons for your answer. |
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Foundations:
| + | [[009C Sample Final 3, Problem 2 Solution|'''<u>Solution</u>''']] |
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| − | |'''1.''' A series <math>\sum a_n</math> is '''absolutely convergent''' if
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| − | | 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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| − | | the series <math>\sum |a_n|</math> diverges and the series <math>\sum a_n</math> converges.
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| − | |}
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| − | '''Solution:''' | + | [[009C Sample Final 3, Problem 2 Detailed Solution|'''<u>Detailed Solution</u>''']] |
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| − | '''(a)'''
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Step 1:
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| − | |First, we take the absolute value of the terms in the original series.
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| − | |Let <math style="vertical-align: -20px">a_n=\frac{(-1)^n}{\sqrt{n}}.</math>
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| − | |Therefore,
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| − | | <math>\begin{array}{rcl}
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| − | \displaystyle{\sum_{n=1}^\infty |a_n|} & = & \displaystyle{\sum_{n=1}^\infty \bigg|\frac{(-1)^n}{\sqrt{n}}\bigg|}\\
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| − | &&\\
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| − | & = & \displaystyle{\sum_{n=1}^\infty \frac{1}{\sqrt{n}}.}
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| − | \end{array}</math>
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| − | |}
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Step 2:
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| − | |This series is a <math style="vertical-align: -5px">p</math>-series with <math style="vertical-align: -14px">p=\frac{1}{2}.</math>
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| − | |Therefore, it diverges.
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| − | |Hence, the series
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| − | | <math>\sum_{n=1}^\infty \frac{(-1)^n}{\sqrt{n}}</math>
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| − | |is not absolutely convergent.
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| − | |}
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| − | '''(b)'''
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Step 1:
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| − | |}
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Step 2:
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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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| − | | '''(a)''' not absolutely convergent
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| − | | '''(b)'''
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| − | |}
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| | [[009C_Sample_Final_3|'''<u>Return to Sample Exam</u>''']] | | [[009C_Sample_Final_3|'''<u>Return to Sample Exam</u>''']] |
