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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Foundations:
| + | [[009C Sample Midterm 1, Problem 4 Solution|'''<u>Solution</u>''']] |
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| − | |'''Direct Comparison Test'''
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| − | | Let <math>\{a_n\}</math> and <math>\{b_n\}</math> be positive sequences where <math style="vertical-align: -3px">a_n\le b_n</math>
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| − | | for all <math style="vertical-align: -3px">n\ge N</math> for some <math style="vertical-align: -3px">N\ge 1.</math>
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| − | | '''1.''' If <math>\sum_{n=1}^\infty b_n</math> converges, then <math>\sum_{n=1}^\infty a_n</math> converges. | |
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| − | | '''2.''' If <math>\sum_{n=1}^\infty a_n</math> diverges, then <math>\sum_{n=1}^\infty b_n</math> diverges.
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| − | '''Solution:''' | + | [[009C Sample Midterm 1, 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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| − | !Step 1:
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| − | |First, we note that
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| − | | <math>\frac{1}{n^23^n}>0</math>
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| − | |for all <math style="vertical-align: -3px">n\ge 1.</math>
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| − | |This means that we can use a comparison test on this series.
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| − | |Let <math style="vertical-align: -14px">a_n=\frac{1}{n^23^n}.</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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| − | |Let <math style="vertical-align: -14px">b_n=\frac{1}{n^2}.</math>
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| − | |We want to compare the series in this problem with
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| − | | <math>\sum_{n=1}^\infty b_n=\sum_{n=1}^\infty \frac{1}{n^2}.</math>
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| − | |This is a <math style="vertical-align: -4px">p</math>-series with <math style="vertical-align: -4px">p=2.</math>
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| − | |Hence, <math>\sum_{n=1}^\infty b_n</math> converges.
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| − | |}
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Step 3:
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| − | |Also, we have <math style="vertical-align: -4px">a_n<b_n</math> since
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| − | | <math>\frac{1}{n^23^n}<\frac{1}{n^2}</math>
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| − | |for all <math style="vertical-align: -3px">n\ge 1.</math>
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| − | |Therefore, the series <math>\sum_{n=1}^\infty a_n</math> converges
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| − | |by the Direct Comparison Test.
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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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| − | | converges (by the Direct Comparison Test)
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| − | |}
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| | [[009C_Sample_Midterm_1|'''<u>Return to Sample Exam</u>''']] | | [[009C_Sample_Midterm_1|'''<u>Return to Sample Exam</u>''']] |
