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

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<span class="exam">Consider the infinite series <math>\sum_{n=2}^\infty 2\bigg(\frac{1}{2^n}-\frac{1}{2^{n+1}}\bigg).</math>
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<span class="exam">Consider the infinite series &nbsp;<math>\sum_{n=2}^\infty 2\bigg(\frac{1}{2^n}-\frac{1}{2^{n+1}}\bigg).</math>
  
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::<span class="exam">a) Find an expression for the <math>n</math>th partial sum <math>s_n</math> of the series.
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<span class="exam">(a) Find an expression for the &nbsp;<math style="vertical-align: 0px">n</math>th partial sum &nbsp;<math style="vertical-align: -3px">s_n</math>&nbsp; of the series.
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::<span class="exam">b) Compute <math>\lim_{n\rightarrow \infty} s_n.</math>
 
  
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<span class="exam">(b) Compute &nbsp;<math style="vertical-align: -11px">\lim_{n\rightarrow \infty} s_n.</math>
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<hr>
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[[009C Sample Midterm 1, Problem 2 Solution|'''<u>Solution</u>''']]
  
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
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!Foundations: &nbsp;
 
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|The <math>n</math>th partial sum, <math>s_n</math> for a series <math>\sum_{n=1}^\infty a_n </math>
 
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|is defined as
 
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&nbsp; &nbsp; &nbsp; &nbsp; <math>s_n=\sum_{i=1}^n a_i.</math>
 
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'''Solution:'''
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[[009C Sample Midterm 1, Problem 2 Detailed Solution|'''<u>Detailed Solution</u>''']]
  
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'''(a)'''
 
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!Step 1: &nbsp;
 
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|We need to find a pattern for the partial sums in order to find a formula.
 
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|We start by calculating <math>s_2</math>. We have
 
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|&nbsp; &nbsp; &nbsp; &nbsp; <math>s_2=2\bigg(\frac{1}{2^2}-\frac{1}{2^3}\bigg).</math>
 
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!Step 2: &nbsp;
 
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|Next, we calculate <math>s_3</math> and <math>s_4.</math> We have
 
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'''(b)'''
 
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!Step 1: &nbsp;
 
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!Step 2: &nbsp;
 
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!Final Answer: &nbsp;
 
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|'''(a)'''
 
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|'''(b)'''
 
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[[009C_Sample_Midterm_1|'''<u>Return to Sample Exam</u>''']]
 
[[009C_Sample_Midterm_1|'''<u>Return to Sample Exam</u>''']]

Latest revision as of 22:52, 11 November 2017

Consider the infinite series  

(a) Find an expression for the  th partial sum    of the series.

(b) Compute  

Solution


Detailed Solution


Return to Sample Exam

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