Difference between revisions of "009C Sample Midterm 2, Problem 5"

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|Let &nbsp;<math style="vertical-align: -5px">a\in (-2R,2R).</math>&nbsp; Then, &nbsp;<math style="vertical-align: -13px">\frac{a}{2} \in (-R,R).</math>&nbsp;
 
|Let &nbsp;<math style="vertical-align: -5px">a\in (-2R,2R).</math>&nbsp; Then, &nbsp;<math style="vertical-align: -13px">\frac{a}{2} \in (-R,R).</math>&nbsp;
 
|-
 
|-
−
|So,  
+
|Since &nbsp;<math style="vertical-align: -19px">\sum_{n=0}^\infty c_nx^n</math>&nbsp; converges in the interval &nbsp;<math style="vertical-align: -5px">(-R,R),</math>&nbsp;
 
|-
 
|-
−
|&nbsp; &nbsp; &nbsp; &nbsp; <math>\sum_{n=0}^\infty c_n\bigg(\frac{a}{2}\bigg)^n</math>
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|&nbsp; <math>\sum_{n=0}^\infty c_n\bigg(\frac{a}{2}\bigg)^n</math>&nbsp; converges.
 
|-
 
|-
−
|converges by assumption.
+
|Since &nbsp;<math style="vertical-align: 0px">a</math>&nbsp; was an arbitrary number in the interval &nbsp;<math style="vertical-align: -5px">(-2R,2R),</math>&nbsp;
−
|-
 
−
|Since &nbsp;<math style="vertical-align: 0px">a</math>&nbsp; was an arbitrary number in the interval &nbsp;<math style="vertical-align: -5px">a\in (-2R,2R),</math>&nbsp;
 
 
|-
 
|-
 
|&nbsp; &nbsp; &nbsp; &nbsp; <math>\sum_{n=0}^\infty c_n\bigg(\frac{x}{2}\bigg)^n</math>
 
|&nbsp; &nbsp; &nbsp; &nbsp; <math>\sum_{n=0}^\infty c_n\bigg(\frac{x}{2}\bigg)^n</math>
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!Step 1: &nbsp;
 
!Step 1: &nbsp;
 
|-
 
|-
−
|
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|Assume that the power series &nbsp;<math style="vertical-align: -19px">\sum_{n=0}^\infty c_nx^n</math>&nbsp; converges.
 +
|-
 +
|Let &nbsp;<math style="vertical-align: 0px">R</math>&nbsp; be the radius of convergence of this power series.
 +
|-
 +
|So, the power series
 +
|-
 +
|&nbsp; &nbsp; &nbsp; &nbsp;<math style="vertical-align: -19px">\sum_{n=0}^\infty c_nx^n</math>&nbsp;
 +
|-
 +
|converges in the interval &nbsp;<math style="vertical-align: -5px">(-R,R).</math>&nbsp;
 
|}
 
|}
  
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!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
|-
 
|-
−
|
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|Let &nbsp;<math style="vertical-align: -5px">a\in (-R,R).</math>&nbsp; Then, &nbsp;<math style="vertical-align: -5px">-a \in (-R,R).</math>&nbsp;
 +
|-
 +
|Since &nbsp;<math style="vertical-align: -19px">\sum_{n=0}^\infty c_nx^n</math>&nbsp; converges in the interval &nbsp;<math style="vertical-align: -5px">(-R,R),</math>&nbsp;
 +
|-
 +
|&nbsp;<math>\sum_{n=0}^\infty c_n(-a)^n</math>&nbsp; converges.
 +
|-
 +
|Since &nbsp;<math style="vertical-align: 0px">a</math>&nbsp; was an arbitrary number in the interval &nbsp;<math style="vertical-align: -5px">(-R,R),</math>&nbsp;
 +
|-
 +
|&nbsp; &nbsp; &nbsp; &nbsp; <math>\sum_{n=0}^\infty c_n(-x)^n</math>
 +
|-
 +
|converges in the interval &nbsp;<math style="vertical-align: -5px">(-R,R).</math>&nbsp;
 
|}
 
|}
  

Revision as of 16:27, 23 April 2017

If    converges, does it follow that the following series converges?

(a)  

(b)  


Foundations:  
If a power series converges, then it has a nonempty interval of convergence.


Solution:

(a)

Step 1:  
Assume that the power series    converges.
Let    be the radius of convergence of this power series.
So, the power series
        
converges in the interval  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (-R,R).}  
Step 2:  
Let  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle a\in (-2R,2R).}   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 \frac{a}{2} \in (-R,R).}  
Since  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 c_nx^n}   converges in 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 (-R,R),}  
  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 c_n\bigg(\frac{a}{2}\bigg)^n}   converges.
Since  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}   was an arbitrary number in 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 (-2R,2R),}  
        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 c_n\bigg(\frac{x}{2}\bigg)^n}
converges in 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 (-2R,2R).}  

(b)

Step 1:  
Assume that the power 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=0}^\infty c_nx^n}   converges.
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 R}   be the radius of convergence of this power series.
So, the power 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=0}^\infty c_nx^n}  
converges in 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 (-R,R).}  
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 a\in (-R,R).}   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 -a \in (-R,R).}  
Since  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 c_nx^n}   converges in 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 (-R,R),}  
 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 c_n(-a)^n}   converges.
Since  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}   was an arbitrary number in 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 (-R,R),}  
        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 c_n(-x)^n}
converges in 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 (-R,R).}  


Final Answer:  
    (a)     converges
    (b)     converges

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