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| | <span class="exam"> Find the radius of convergence and interval of convergence of the series. | | <span class="exam"> Find the radius of convergence and interval of convergence of the series. |
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| − | ::<span class="exam">a) <math>\sum_{n=0}^\infty n^nx^n</math>
| + | <span class="exam">(a) <math>\sum_{n=1}^\infty n^nx^n</math> |
| − | ::<span class="exam">b) <math>\sum_{n=0}^\infty \frac{(x+1)^n}{\sqrt{n}}</math>
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| | + | <span class="exam">(b) <math>\sum_{n=1}^\infty \frac{(x+1)^n}{\sqrt{n}}</math> |
| | + | <hr> |
| | + | [[009C Sample Midterm 2, Problem 4 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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| − | | Root Test
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| − | | Ratio Test
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| − | |}
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| | + | [[009C Sample Midterm 2, Problem 4 Detailed Solution|'''<u>Detailed Solution</u>''']] |
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| − | '''Solution:'''
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| − | '''(a)'''
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Step 1:
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| − | |We begin by applying the Root Test.
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| − | |We have
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| − | <math>\begin{array}{rcl}
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| − | \displaystyle{\lim_{n\rightarrow \infty} \sqrt{|a_n|}} & = & \displaystyle{\lim_{n\rightarrow \infty} \sqrt{|n^nx^n|}}\\
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| − | &&\\
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| − | & = & \displaystyle{\lim_{n\rightarrow \infty} |n^nx^n|^{\frac{1}{n}}}\\
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| − | &&\\
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| − | & = & \displaystyle{\lim_{n\rightarrow \infty} |nx|}\\
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| − | &&\\
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| − | & = & \displaystyle{n|x|}\\
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| − | &&\\
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| − | & = & \displaystyle{|x|\lim_{n\rightarrow \infty} n}\\
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| − | &&\\
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| − | & = & \displaystyle{\infty}
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| − | \end{array}</math>
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| − | |}
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Step 2:
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| − | |This means that as long as <math>x\ne 0,</math> this series diverges.
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| − | |Hence, the radius of convergence is <math>R=0</math> and
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| − | |-
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| − | |the interval of convergence is <math>\{0\}.</math>
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| − | |}
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| − | '''(b)'''
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Step 1:
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| − | |}
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Step 2:
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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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| − | | '''(a)''' The radius of convergence is <math>R=0</math> and the interval of convergence is <math>\{0\}.</math>
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| − | |-
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| − | | '''(b)''' The radius of convergence is <math>R=1</math> and the interval fo convergence is <math>(2,4].</math>
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| − | |}
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| | [[009C_Sample_Midterm_2|'''<u>Return to Sample Exam</u>''']] | | [[009C_Sample_Midterm_2|'''<u>Return to Sample Exam</u>''']] |
Find the radius of convergence and interval of convergence of the series.
(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 \sum_{n=1}^\infty \frac{(x+1)^n}{\sqrt{n}}}
Solution
Detailed Solution
Return to Sample Exam