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| | ::<math>\sum_{n=1}^{\infty} \frac{n^3+7n}{\sqrt{1+n^{10}}}</math> | | ::<math>\sum_{n=1}^{\infty} \frac{n^3+7n}{\sqrt{1+n^{10}}}</math> |
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
| + | <hr> |
| − | !Foundations:
| + | [[009C Sample Final 3, Problem 3 Solution|'''<u>Solution</u>''']] |
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| − | |'''Limit Comparison Test''' | |
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| − | | Let <math>\{a_n\}</math> and <math>\{b_n\}</math> be positive sequences.
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| − | | If <math>\lim_{n\rightarrow \infty} \frac{a_n}{b_n}=L,</math> where <math>L</math> is a positive real number,
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| − | |-
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| − | | then <math>\sum_{n=1}^\infty a_n</math> and <math>\sum_{n=1}^\infty b_n</math> either both converge or both diverge.
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| − | |}
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| − | '''Solution:''' | + | [[009C Sample Final 3, Problem 3 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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| − | |-
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| − | |First, we note that
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| − | | <math>\frac{n^3+7n}{\sqrt{1+n^{10}}}>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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| − | |-
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| − | |Let <math style="vertical-align: -14px">a_n=\frac{n^3+7n}{\sqrt{1+n^{10}}}.</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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| − | |}
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Step 3:
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| − | |Now, we have
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| − | | <math>\begin{array}{rcl}
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| − | \displaystyle{\lim_{n\rightarrow \infty} \frac{a_n}{b_n}} & = & \displaystyle{\lim_{n\rightarrow \infty} \frac{(\frac{n^3+7n}{\sqrt{1+n^{10}}})}{(\frac{1}{n^2})}}\\
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| − | &&\\
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| − | & = & \displaystyle{\lim_{n\rightarrow \infty} \frac{n^5+7n^3}{\sqrt{1+n^{10}}}}\\
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| − | &&\\
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| − | & = & \displaystyle{\lim_{n\rightarrow \infty} \frac{n^5+7n^3}{\sqrt{1+n^{10}}} \bigg(\frac{\frac{1}{n^5}}{\frac{1}{n^5}}\bigg)}\\
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| − | &&\\
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| − | & = & \displaystyle{\lim_{n\rightarrow \infty} \frac{1+\frac{7}{n^4}}{\sqrt{\frac{1}{n^{10}}+1}}}\\
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| − | &&\\
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| − | & = & \displaystyle{1.}
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| − | \end{array}</math>
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| − | |Therefore, the series
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| − | | <math>\sum_{n=1}^{\infty} \frac{n^3+7n}{\sqrt{1+n^{10}}}</math>
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| − | |-
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| − | |converges by the Limit 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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| − | |-
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| − | | converges
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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>''']] |
