Difference between revisions of "009C Sample Midterm 3, Problem 5"

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<span class="exam">Find the radius of convergence and the interval of convergence
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<span class="exam"> Find the radius of convergence and the interval of convergence
 
of the series.  
 
of the series.  
  
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::<span class="exam">(a) (6 points) &nbsp;&nbsp;&nbsp;&nbsp; <math>{\displaystyle \sum_{n=0}^{\infty}}\frac{(-1)^{n}x^{n}}{n+1}.</math>
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<span class="exam">(a) &nbsp;<math>{\displaystyle \sum_{n=0}^{\infty}}\frac{(-1)^{n}x^{n}}{n+1}</math>
  
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::<span class="exam">(b) (6 points) &nbsp;&nbsp;&nbsp;&nbsp; <math>{\displaystyle \sum_{n=0}^{\infty}}\frac{(x+1)^{n}}{n^{2}}.</math>
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<span class="exam">(b) &nbsp;<math>{\displaystyle \sum_{n=0}^{\infty}}\frac{(x+1)^{n}}{n^{2}}</math>
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<hr>
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[[009C Sample Midterm 3, Problem 5 Solution|'''<u>Solution</u>''']]
  
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! Foundations: &nbsp;
 
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|When we are asked to find the radius of convergence, we are given a series where
 
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::<math>a_n=f(x-c)\cdot g(n)</math>
 
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|where <math style="vertical-align: -20%">f</math> and <math style="vertical-align: -20%">g</math> are functions of <math style="vertical-align: 0%">x</math> and <math style="vertical-align: 0%">n</math> respectively, and <math style="vertical-align: 0%">c</math> is a constant (frequently zero).  We need to find a bound on <math style="vertical-align: -22%">|x-c|</math> such that whenever <math style="vertical-align: -22%">|x-c|<L</math>, the ratio test
 
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::<math>\left|\frac{a_{n+1}}{a_n}\right|</math>
 
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|is satisfied.  When we do, the interval will be <math style="vertical-align: -20%">(c-L,c+L)</math>.  However, the boundary values for <math style="vertical-align: 0%">x</math>, <math style="vertical-align: 0%">c-L</math> and <math style="vertical-align: -8%">c+L</math> must be tested individually for convergence.  Most often, one will produce an alternating, convergent series while the other will produce a divergent, non-alternating series.
 
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&nbsp;'''Solution:'''
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[[009C Sample Midterm 3, Problem 5 Detailed Solution|'''<u>Detailed Solution</u>''']]
  
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!(b): &nbsp;
 
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
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!Final Answer: &nbsp;
 
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[[009C_Sample_Midterm_3|'''<u>Return to Sample Exam</u>''']]
 
[[009C_Sample_Midterm_3|'''<u>Return to Sample Exam</u>''']]

Latest revision as of 09:47, 28 November 2017

Find the radius of convergence and the interval of convergence of the series.

(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 {\displaystyle \sum_{n=0}^{\infty}}\frac{(-1)^{n}x^{n}}{n+1}}

(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 {\displaystyle \sum_{n=0}^{\infty}}\frac{(x+1)^{n}}{n^{2}}}

Solution


Detailed Solution


Return to Sample Exam

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