Difference between revisions of "009C Sample Midterm 2, Problem 1"

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|Sum formula for geometric series
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!Step 1:    
 
!Step 1:    
 
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|First, we not that this is a geometric series with <math>r=\frac{1}{4}.</math>
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|Since <math>|r|=\frac{1}{4}<1,</math>
 
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|this series converges.
 
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!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
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|Now, we need to find the sum of this series.
 
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|The first term of the series is <math>a_1=\frac{1}{2}.</math>
 
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|Hence, the sum of the series is
 
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&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
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\displaystyle{\frac{a_1}{1-r}} & = & \displaystyle{\frac{\frac{1}{2}}{1-\frac{1}{4}}}\\
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&&\\
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& = & \displaystyle{\frac{\big(\frac{1}{2}\big)}{\big(\frac{3}{4}\big)}}\\
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&&\\
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& = & \displaystyle{\frac{2}{3}}
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\end{array}</math>
 
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|'''(a)'''  
 
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|&nbsp; &nbsp; '''(b)''' &nbsp; &nbsp; <math>\frac{2}{3}</math>
 
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[[009C_Sample_Midterm_2|'''<u>Return to Sample Exam</u>''']]
 
[[009C_Sample_Midterm_2|'''<u>Return to Sample Exam</u>''']]

Revision as of 10:13, 13 February 2017

Evaluate:

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 \lim _{n\rightarrow \infty} \frac{1}{\big(\frac{n-4}{n}\big)^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=1}^\infty \frac{1}{2} \bigg(\frac{1}{4}\bigg)^{n-1} }


Foundations:  
Sum formula for geometric series

Solution:

(a)

Step 1:  
Step 2:  

(b)

Step 1:  
First, we not that this is a geometric series with 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{1}{4}.}
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|=\frac{1}{4}<1,}
this series converges.
Step 2:  
Now, we need to find the sum of this series.
The first term of the 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 a_1=\frac{1}{2}.}
Hence, the sum of the 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 \begin{array}{rcl} \displaystyle{\frac{a_1}{1-r}} & = & \displaystyle{\frac{\frac{1}{2}}{1-\frac{1}{4}}}\\ &&\\ & = & \displaystyle{\frac{\big(\frac{1}{2}\big)}{\big(\frac{3}{4}\big)}}\\ &&\\ & = & \displaystyle{\frac{2}{3}} \end{array}}

Final Answer:  
(a)
    (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 \frac{2}{3}}

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