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

From Grad Wiki
Jump to navigation Jump to search
Line 1: Line 1:
−
<span class="exam">If <math>\sum_{n=0}^\infty c_nx^n</math> converges, does it follow that the following series converges?
+
<span class="exam">If &nbsp;<math>\sum_{n=0}^\infty c_nx^n</math>&nbsp; converges, does it follow that the following series converges?
  
−
<span class="exam">(a) <math>\sum_{n=0}^\infty c_n\bigg(\frac{x}{2}\bigg)^n</math>
+
<span class="exam">(a) &nbsp;<math>\sum_{n=0}^\infty c_n\bigg(\frac{x}{2}\bigg)^n</math>
  
−
<span class="exam">(b) <math>\sum_{n=0}^\infty c_n(-x)^n </math>
+
<span class="exam">(b) &nbsp;<math>\sum_{n=0}^\infty c_n(-x)^n </math>
  
  
Line 10: Line 10:
 
!Foundations: &nbsp;  
 
!Foundations: &nbsp;  
 
|-
 
|-
−
|A geometric series <math>\sum_{n=0}^{\infty} ar^n</math> converges if <math style="vertical-align: -6px">|r|<1.</math>
+
|A geometric series &nbsp;<math>\sum_{n=0}^{\infty} ar^n</math>&nbsp; converges if &nbsp;<math style="vertical-align: -6px">|r|<1.</math>
 
|}
 
|}
  
Line 20: Line 20:
 
!Step 1: &nbsp;
 
!Step 1: &nbsp;
 
|-
 
|-
−
|First, we notice that <math>\sum_{n=0}^\infty c_nx^n</math> is a geometric series.
+
|First, we notice that &nbsp;<math>\sum_{n=0}^\infty c_nx^n</math>&nbsp; is a geometric series.
 
|-
 
|-
−
|We have <math style="vertical-align: -1px">r=x.</math>
+
|We have &nbsp;<math style="vertical-align: -1px">r=x.</math>
 
|-
 
|-
 
|Since this series converges,
 
|Since this series converges,
Line 32: Line 32:
 
!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
|-
 
|-
−
|The series <math>\sum_{n=0} c_n\bigg(\frac{x}{2}\bigg)^n</math> is also a geometric series.
+
|The series &nbsp;<math>\sum_{n=0} c_n\bigg(\frac{x}{2}\bigg)^n</math>&nbsp; is also a geometric series.
 
|-
 
|-
−
|For this series, <math style="vertical-align: -13px">r=\frac{x}{2}.</math>
+
|For this series, &nbsp;<math style="vertical-align: -13px">r=\frac{x}{2}.</math>
 
|-
 
|-
 
|Now, we notice
 
|Now, we notice
Line 47: Line 47:
 
\end{array}</math>
 
\end{array}</math>
 
|-
 
|-
−
|since <math style="vertical-align: -5px">|x|<1.</math>  
+
|since &nbsp;<math style="vertical-align: -5px">|x|<1.</math>  
 
|-
 
|-
−
| Since <math style="vertical-align: -5px">|r|<1,</math> this series converges.
+
| Since &nbsp;<math style="vertical-align: -5px">|r|<1,</math>&nbsp; this series converges.
 
|}
 
|}
  
Line 56: Line 56:
 
!Step 1: &nbsp;
 
!Step 1: &nbsp;
 
|-
 
|-
−
|First, we notice that <math>\sum_{n=0}^\infty c_nx^n</math> is a geometric series.
+
|First, we notice that &nbsp;<math>\sum_{n=0}^\infty c_nx^n</math>&nbsp; is a geometric series.
 
|-
 
|-
−
|We have <math style="vertical-align: -1px">r=x.</math>
+
|We have &nbsp;<math style="vertical-align: -1px">r=x.</math>
 
|-
 
|-
 
|Since this series converges,
 
|Since this series converges,
Line 68: Line 68:
 
!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
|-
 
|-
−
|The series <math>\sum_{n=0}^\infty c_n(-x)^n</math> is also a geometric series.
+
|The series &nbsp;<math>\sum_{n=0}^\infty c_n(-x)^n</math>&nbsp; is also a geometric series.
 
|-
 
|-
−
|For this series, <math style="vertical-align: -1px">r=-x.</math>
+
|For this series, &nbsp;<math style="vertical-align: -1px">r=-x.</math>
 
|-
 
|-
 
|Now, we notice
 
|Now, we notice
Line 83: Line 83:
 
\end{array}</math>
 
\end{array}</math>
 
|-
 
|-
−
|since <math style="vertical-align: -5px">|x|<1.</math>  
+
|since &nbsp;<math style="vertical-align: -5px">|x|<1.</math>  
 
|-
 
|-
−
|Since <math style="vertical-align: -5px">|r|<1,</math> this series converges.
+
|Since &nbsp;<math style="vertical-align: -5px">|r|<1,</math>&nbsp; this series converges.
 
|}
 
|}
  

Revision as of 18:14, 26 February 2017

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 \sum_{n=0}^\infty c_nx^n}   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:  
A geometric 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} ar^n}   converges 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 |r|<1.}


Solution:

(a)

Step 1:  
First, we notice 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 \sum_{n=0}^\infty c_nx^n}   is a geometric series.
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 r=x.}
Since this series converges,
        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|=|x|<1.}
Step 2:  
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=0} c_n\bigg(\frac{x}{2}\bigg)^n}   is also a geometric series.
For this 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 r=\frac{x}{2}.}
Now, we notice

        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{|r|} & = & \displaystyle{\bigg|\frac{x}{2}\bigg|}\\ &&\\ & = & \displaystyle{\frac{|x|}{2}}\\ &&\\ & < & \displaystyle{\frac{1}{2}} \end{array}}

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 |x|<1.}
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 |r|<1,}   this series converges.

(b)

Step 1:  
First, we notice 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 \sum_{n=0}^\infty c_nx^n}   is a geometric series.
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 r=x.}
Since this series converges,
        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|=|x|<1.}
Step 2:  
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=0}^\infty c_n(-x)^n}   is also a geometric series.
For this 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 r=-x.}
Now, we notice

        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{|r|} & = & \displaystyle{|-x|}\\ &&\\ & = & \displaystyle{|x|}\\ &&\\ & < & \displaystyle{1} \end{array}}

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 |x|<1.}
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 |r|<1,}   this series converges.


Final Answer:  
    (a)     The series converges.
    (b)     The series converges.

Return to Sample Exam

Retrieved from "https://gradwiki.math.ucr.edu/index.php?title=009C_Sample_Midterm_2,_Problem_5&oldid=2833"