Difference between revisions of "009C Sample Midterm 2, Problem 4"

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!Foundations:    
 
!Foundations:    
 
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| Root Test
 
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| Ratio Test
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'''Solution:'''
 
'''Solution:'''
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'''(a)'''
 
'''(a)'''
 
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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!Step 1:    
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!Step 1:  
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|-
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|We begin by applying the Root Test.
 
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|We have
 
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&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
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\displaystyle{\lim_{n\rightarrow \infty} \sqrt{|a_n|}} & = & \displaystyle{\lim_{n\rightarrow \infty} \sqrt{|n^nx^n|}}\\
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&&\\
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& = & \displaystyle{\lim_{n\rightarrow \infty} |n^nx^n|^{\frac{1}{n}}}\\
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&&\\
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& = & \displaystyle{\lim_{n\rightarrow \infty} |nx|}\\
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&&\\
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& = & \displaystyle{n|x|}\\
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&&\\
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& = & \displaystyle{|x|\lim_{n\rightarrow \infty} n}\\
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&&\\
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& = & \displaystyle{\infty}
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\end{array}</math>
 
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!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
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|This means that as long as <math>x\ne 0,</math> this series diverges.
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|Hence, the radius of convergence is <math>R=0</math> and
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|the interval of convergence is <math>\{0\}.</math>
 
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"

Revision as of 10:17, 13 February 2017

Find the radius of convergence and interval of convergence of the series.

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 \sum_{n=0}^\infty n^nx^n}
b) 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=0}^\infty \frac{(x+1)^n}{\sqrt{n}}}


Foundations:  
Root Test
Ratio Test


Solution:

(a)

Step 1:  
We begin by applying the Root Test.
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} \sqrt{|a_n|}} & = & \displaystyle{\lim_{n\rightarrow \infty} \sqrt{|n^nx^n|}}\\ &&\\ & = & \displaystyle{\lim_{n\rightarrow \infty} |n^nx^n|^{\frac{1}{n}}}\\ &&\\ & = & \displaystyle{\lim_{n\rightarrow \infty} |nx|}\\ &&\\ & = & \displaystyle{n|x|}\\ &&\\ & = & \displaystyle{|x|\lim_{n\rightarrow \infty} n}\\ &&\\ & = & \displaystyle{\infty} \end{array}}

Step 2:  
This means that as long as 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 x\ne 0,} this series diverges.
Hence, the radius of convergence is 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 R=0} and
the interval of convergence is 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 \{0\}.}

(b)

Step 1:  
Step 2:  


Final Answer:  
(a)
(b)

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