Difference between revisions of "009C Sample Final 3, Problem 3"

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!Foundations:    
 
!Foundations:    
 
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|'''Limit Comparison Test'''
 
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|&nbsp; &nbsp; &nbsp; &nbsp; Let &nbsp;<math>\{a_n\}</math>&nbsp; and &nbsp;<math>\{b_n\}</math>&nbsp; be positive sequences.
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|&nbsp; &nbsp; &nbsp; &nbsp; If &nbsp;<math>\lim_{n\rightarrow \infty} \frac{a_n}{b_n}=L,</math>&nbsp; where &nbsp;<math>L</math> &nbsp; is a positive real number,
 
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|&nbsp; &nbsp; &nbsp; &nbsp; then &nbsp;<math>\sum_{n=1}^\infty a_n</math>&nbsp; and &nbsp;<math>\sum_{n=1}^\infty b_n</math>&nbsp; either both converge or both diverge.
 
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!Step 1: &nbsp;  
 
!Step 1: &nbsp;  
 
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|First, we note that
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|&nbsp; &nbsp; &nbsp; &nbsp; <math>\frac{n^3+7n}{\sqrt{1+n^{10}}}>0</math>
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|for all &nbsp;<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 &nbsp;<math style="vertical-align: -14px">a_n=\frac{n^3+7n}{\sqrt{1+n^{10}}}.</math>
 
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Step 2: &nbsp;
 
!Step 2: &nbsp;
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|Let &nbsp;<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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|&nbsp; &nbsp; &nbsp; &nbsp; <math>\sum_{n=1}^\infty b_n=\sum_{n=1}^\infty \frac{1}{n^2}.</math>
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|This is a &nbsp;<math style="vertical-align: -4px">p</math>-series with &nbsp;<math style="vertical-align: -4px">p=2.</math>
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|Hence, &nbsp;<math>\sum_{n=1}^\infty b_n</math>&nbsp; converges
 
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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!Step 3: &nbsp;
 
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|Now, we have
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|&nbsp; &nbsp; &nbsp; &nbsp; <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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|&nbsp; &nbsp; &nbsp; &nbsp; <math>\sum_{n=1}^{\infty} \frac{n^3+7n}{\sqrt{1+n^{10}}}</math>
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|converges by the Limit Comparison Test.
 
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!Final Answer: &nbsp;  
 
!Final Answer: &nbsp;  
 
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|&nbsp; &nbsp; &nbsp; &nbsp; converges
 
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[[009C_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]
 
[[009C_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]

Revision as of 15:43, 5 March 2017

Test if the following series converges or diverges. Give reasons and clearly state if you are using any standard test.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=1}^{\infty} \frac{n^3+7n}{\sqrt{1+n^{10}}}}
Foundations:  
Limit Comparison Test
        Let  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{a_n\}}   and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{b_n\}}   be positive sequences.
        If  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n\rightarrow \infty} \frac{a_n}{b_n}=L,}   where  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L}   is a positive real number,
        then  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=1}^\infty a_n}   and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=1}^\infty b_n}   either both converge or both diverge.


Solution:

Step 1:  
First, we note that
        Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{n^3+7n}{\sqrt{1+n^{10}}}>0}
for all  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n\ge 1.}
This means that we can use a comparison test on this series.
Let  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_n=\frac{n^3+7n}{\sqrt{1+n^{10}}}.}
Step 2:  
Let  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b_n=\frac{1}{n^2}.}
We want to compare the series in this problem with
        Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=1}^\infty b_n=\sum_{n=1}^\infty \frac{1}{n^2}.}
This is a  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} -series with  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p=2.}
Hence,  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=1}^\infty b_n}   converges
Step 3:  
Now, we have
        Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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}}
Therefore, the series
        Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=1}^{\infty} \frac{n^3+7n}{\sqrt{1+n^{10}}}}
converges by the Limit Comparison Test.


Final Answer:  
        converges

Return to Sample Exam

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