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| | <span class="exam">Determine convergence or divergence: | | <span class="exam">Determine convergence or divergence: |
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| − | ::<span class="exam">a) <math>\sum_{n=1}^\infty (-1)^n \sqrt{\frac{1}{n}}</math>
| + | <span class="exam">(a) <math>\sum_{n=1}^\infty (-1)^n \sqrt{\frac{1}{n}}</math> |
| − | ::<span class="exam">b) <math>\sum_{n=1}^\infty (-2)^n \frac{n!}{n^n} </math>
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| | + | <span class="exam">(b) <math>\sum_{n=1}^\infty (-2)^n \frac{n!}{n^n} </math> |
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
| + | <hr> |
| − | !Foundations:
| + | [[009C Sample Midterm 2, Problem 3 Solution|'''<u>Solution</u>''']] |
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| − | |Alternating Series Test
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| − | |Ratio Test
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| − | '''Solution:'''
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| − | '''(a)''' | + | [[009C Sample Midterm 2, Problem 3 Detailed Solution|'''<u>Detailed Solution</u>''']] |
| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Step 1:
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| − | |First, we have
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| − | | <math>\sum_{n=1}^\infty (-1)^n \sqrt{\frac{1}{n}}=\sum_{n=1}^\infty (-1)^n \frac{1}{\sqrt{n}}.</math>
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Step 2:
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| − | |We notice that the series is alternating.
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| − | |Let <math> b_n=\frac{1}{\sqrt{n}}.</math>
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| − | |The sequence <math>\{b_n\}</math> is decreasing since
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| − | | <math>\frac{1}{\sqrt{n+1}}<\frac{1}{\sqrt{n}}</math>
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| − | |for all <math>n\ge 1.</math>
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| − | |Also,
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| − | | <math>\lim_{n\rightarrow \infty}b_n=\lim_{n\rightarrow \infty}\frac{1}{\sqrt{n}}=0.</math>
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| − | |Therefore, the series <math>\sum_{n=1}^\infty \frac{(-1)^n}{\sqrt{n}}</math> converges by the Alternating Series Test.
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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)''' converges
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| − | | '''(b)''' converges
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| − | |}
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| | [[009C_Sample_Midterm_2|'''<u>Return to Sample Exam</u>''']] | | [[009C_Sample_Midterm_2|'''<u>Return to Sample Exam</u>''']] |
