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;
 
|-
 
|-
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|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;
 
|-
 
|-
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|&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.
 
|-
 
|-
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|converges by assumption.
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|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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|
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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)  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}

(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 c_n(-x)^n }


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


Solution:

(a)

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 (-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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