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2017-03-19T18:43:00Z

Created page with "<span class="exam">Test if the following series converges or diverges. Give reasons and clearly state if you are using any standard test. ::<math>\sum_{n=1}^{\infty} \frac{n^..."

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<span class="exam">Test if the following series converges or diverges. Give reasons and clearly state if you are using any standard test.

::<math>\sum_{n=1}^{\infty} \frac{n^3+7n}{\sqrt{1+n^{10}}}</math>

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Foundations:  
|-
|'''Limit Comparison Test'''
|-
|        Let  <math>\{a_n\}</math>  and  <math>\{b_n\}</math>  be positive sequences.
|-
|        If  <math style="vertical-align: -17px">\lim_{n\rightarrow \infty} \frac{a_n}{b_n}=L,</math>  where  <math style="vertical-align: -1px">L</math>  is a positive real number,
|-
|        then  <math style="vertical-align: -20px">\sum_{n=1}^\infty a_n</math>  and  <math style="vertical-align: -20px">\sum_{n=1}^\infty b_n</math>  either both converge or both diverge.
|}

'''Solution:'''

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 1:  
|-
|First, we note that
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|        <math>\frac{n^3+7n}{\sqrt{1+n^{10}}}>0</math>
|-
|for all  <math style="vertical-align: -3px">n\ge 1.</math>
|-
|This means that we can use a comparison test on this series.
|-
|Let  <math style="vertical-align: -19px">a_n=\frac{n^3+7n}{\sqrt{1+n^{10}}}.</math>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 2:  
|-
|Let  <math style="vertical-align: -14px">b_n=\frac{1}{n^2}.</math>
|-
|We want to compare the series in this problem with
|-
|        <math>\sum_{n=1}^\infty b_n=\sum_{n=1}^\infty \frac{1}{n^2}.</math>
|-
|This is a  <math style="vertical-align: -4px">p</math>-series with  <math style="vertical-align: -4px">p=2.</math>
|-
|Hence,  <math>\sum_{n=1}^\infty b_n</math>  converges
|-
|
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 3:  
|-
|Now, we have
|-
|        <math>\begin{array}{rcl}
\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})}}\\
&&\\
& = & \displaystyle{\lim_{n\rightarrow \infty} \frac{n^5+7n^3}{\sqrt{1+n^{10}}}}\\
&&\\
& = & \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)}\\
&&\\
& = & \displaystyle{\lim_{n\rightarrow \infty} \frac{1+\frac{7}{n^4}}{\sqrt{\frac{1}{n^{10}}+1}}}\\
&&\\
& = & \displaystyle{1.}
\end{array}</math>
|-
|Therefore, the series
|-
|        <math>\sum_{n=1}^{\infty} \frac{n^3+7n}{\sqrt{1+n^{10}}}</math>
|-
|converges by the Limit Comparison Test.
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Final Answer:  
|-
|        converges (by the Limit Comparison Test)
|}
[[009C_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]

009C Sample Final 3, Problem 3 - Revision history

MathAdmin: Created page with "Test if the following series converges or diverges. Give reasons and clearly state if you are using any standard test. ::\sum_{n=1}^{\infty} \frac{n^..."

MathAdmin: Created page with "Test if the following series converges or diverges. Give reasons and clearly state if you are using any standard test. :: $\sum_{n=1}^{\infty} \frac{n^..."$