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| − | <span class="exam">Find the radius of convergence and the interval of convergence | + | <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) <math>{\displaystyle \sum_{n=0}^{\infty}}\frac{(-1)^{n}x^{n}}{n+1}.</math>
| + | <span class="exam">(a) <math>{\displaystyle \sum_{n=0}^{\infty}}\frac{(-1)^{n}x^{n}}{n+1}</math> |
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| − | ::<span class="exam">(b) (6 points) <math>{\displaystyle \sum_{n=0}^{\infty}}\frac{(x+1)^{n}}{n^{2}}.</math>
| + | <span class="exam">(b) <math>{\displaystyle \sum_{n=0}^{\infty}}\frac{(x+1)^{n}}{n^{2}}</math> |
| | + | <hr> |
| | + | [[009C Sample Midterm 3, Problem 5 Solution|'''<u>Solution</u>''']] |
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | ! Foundations:
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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 (radius) on <math style="vertical-align: -22%">|x-c|</math> such that whenever <math style="vertical-align: -22%">|x-c|<r</math>, the ratio test
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| − | ::<math>\left|\frac{a_{n+1}}{a_n}\right|<1.</math>
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| − | When we do, the interval will be <math style="vertical-align: -20%">(c-r,c+r)</math>. However, the boundary values for <math style="vertical-align: 0%">x</math>, <math style="vertical-align: 0%">c-r</math> and <math style="vertical-align: -8%">c+r</math> must be tested individually for convergence. Many times, one boundary value will produce an alternating, convergent series while the other will produce a divergent, non-alternating series. As a result, intervals of convergence may not be strictly open.
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| − | |}
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| − | '''Solution:'''
| + | [[009C Sample Midterm 3, Problem 5 Detailed Solution|'''<u>Detailed Solution</u>''']] |
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !(a):
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| − | |}
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !(b):
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| − | |}
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Final Answer:
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| − | |}
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| | [[009C_Sample_Midterm_3|'''<u>Return to Sample Exam</u>''']] | | [[009C_Sample_Midterm_3|'''<u>Return to Sample Exam</u>''']] |
