Difference between revisions of "009C Sample Final 1, Problem 5"

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!Step 2:  
 
!Step 2:  
 
|-
 
|-
−
|Thus, we have <math>|x|<1</math> and the radius of convergence of this series is <math>1</math>.
+
|Thus, we have <math style="vertical-align: -5px">|x|<1</math> and the radius of convergence of this series is <math style="vertical-align: -1px">1</math>.
 
|}
 
|}
  
Line 47: Line 47:
 
!Step 1: &nbsp;  
 
!Step 1: &nbsp;  
 
|-
 
|-
−
|From part (a), we know the series converges inside the interval <math>(-1,1)</math>.
+
|From part (a), we know the series converges inside the interval <math style="vertical-align: -5px">(-1,1)</math>.
 
|-
 
|-
 
|Now, we need to check the endpoints of the interval for convergence.
 
|Now, we need to check the endpoints of the interval for convergence.
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!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
|-
 
|-
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|For <math>x=1</math>, the series becomes <math>\sum_{n=1}^{\infty}n</math>, which diverges by the Divergence Test.
+
|For <math style="vertical-align: -2px">x=1</math>, the series becomes <math>\sum_{n=1}^{\infty}n</math>, which diverges by the Divergence Test.
 
|}
 
|}
  
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!Step 3: &nbsp;
 
!Step 3: &nbsp;
 
|-
 
|-
−
|For <math>x=-1</math>, the series becomes <math>\sum_{n=1}^{\infty}(-1)^n n</math>, which diverges by the Divergence Test.
+
|For <math style="vertical-align: -2px">x=-1</math>, the series becomes <math>\sum_{n=1}^{\infty}(-1)^n n</math>, which diverges by the Divergence Test.
 
|-
 
|-
−
|Thus, the interval of convergence is <math>(-1,1)</math>.
+
|Thus, the interval of convergence is <math style="vertical-align: -5px">(-1,1)</math>.
 
|}
 
|}
  
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!Step 1: &nbsp;  
 
!Step 1: &nbsp;  
 
|-
 
|-
−
|Recall we have the geometric series formula <math>\frac{1}{1-x}=\sum_{n=0}^{\infty} x^n</math> for <math>|x|<1</math>.  
+
|Recall that we have the geometric series formula <math>\frac{1}{1-x}=\sum_{n=0}^{\infty} x^n</math> for <math>|x|<1</math>.  
 
|-
 
|-
 
|Now, we take the derivative of both sides of the last equation to get
 
|Now, we take the derivative of both sides of the last equation to get
 
|-
 
|-
−
|<math>\frac{1}{(1-x)^2}=\sum_{n=1}^{\infty}nx^{n-1}</math>.
+
|
 +
::<math>\frac{1}{(1-x)^2}=\sum_{n=1}^{\infty}nx^{n-1}</math>.
 
|}
 
|}
  
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!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
|-
 
|-
−
|Now, we multiply the last equation in Step 1 by <math>x</math>.  
+
|Now, we multiply the last equation in Step 1 by <math style="vertical-align: 0px">x</math>.  
 
|-
 
|-
 
|So, we have <math>\frac{x}{(1-x)^2}=\sum_{n=1}^{\infty}nx^{n}=f(x)</math>.
 
|So, we have <math>\frac{x}{(1-x)^2}=\sum_{n=1}^{\infty}nx^{n}=f(x)</math>.
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!Final Answer: &nbsp;  
 
!Final Answer: &nbsp;  
 
|-
 
|-
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|'''(a)''' <math>1</math>
+
|'''(a)''' <math style="vertical-align: -3px">1</math>
 
|-
 
|-
−
|'''(b)''' <math>(-1,1)</math>
+
|'''(b)''' <math style="vertical-align: -3px">(-1,1)</math>
 
|-
 
|-
−
|'''(c)''' <math>f(x)=\frac{x}{(1-x)^2}</math>
+
|'''(c)''' <math style="vertical-align: -18px">f(x)=\frac{x}{(1-x)^2}</math>
 
|}
 
|}
 
[[009C_Sample_Final_1|'''<u>Return to Sample Exam</u>''']]
 
[[009C_Sample_Final_1|'''<u>Return to Sample Exam</u>''']]

Revision as of 14:00, 22 February 2016

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 f(x)=\sum_{n=1}^{\infty} nx^n}

a) Find the radius of convergence of the power series.

b) Determine the interval of convergence of the power series.

c) Obtain an explicit formula for the function .

Foundations:  
Review ratio test.

Solution:

(a)

Step 1:  
To find the radius of convergence, we use the ratio 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}\bigg|\frac{a_{n+1}}{a_n}\bigg|} & = & \displaystyle{\lim_{n \rightarrow \infty}\bigg|\frac{(n+1)x^{n+1}}{nx^n}}\bigg|\\ &&\\ & = & \displaystyle{\lim_{n \rightarrow \infty}\bigg|\frac{n+1}{n}x\bigg|}\\ &&\\ & = & \displaystyle{|x|\lim_{n \rightarrow \infty}\frac{n+1}{n}}\\ &&\\ & = & \displaystyle{|x|}\\ \end{array}}
Step 2:  
Thus, 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 |x|<1} and the radius of convergence of this series 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 1} .

(b)

Step 1:  
From part (a), we know the series converges inside the interval 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 (-1,1)} .
Now, we need to check the endpoints of the interval for convergence.
Step 2:  
For 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=1} , the series becomes 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}n} , which diverges by the Divergence Test.
Step 3:  
For 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=-1} , the series becomes 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}(-1)^n n} , which diverges by the Divergence Test.
Thus, 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 (-1,1)} .

(c)

Step 1:  
Recall that we have the geometric series formula 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{1}{1-x}=\sum_{n=0}^{\infty} x^n} for 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|<1} .
Now, we take the derivative of both sides of the last equation to get
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{1}{(1-x)^2}=\sum_{n=1}^{\infty}nx^{n-1}} .
Step 2:  
Now, we multiply the last equation in Step 1 by 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} .
So, 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 \frac{x}{(1-x)^2}=\sum_{n=1}^{\infty}nx^{n}=f(x)} .
Thus, 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 f(x)=\frac{x}{(1-x)^2}} .
Final Answer:  
(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 1}
(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 (-1,1)}
(c) 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 f(x)=\frac{x}{(1-x)^2}}

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