Difference between revisions of "009C Sample Midterm 2, Problem 4"

Latest revision as of 12:38, 12 November 2017

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

(a) $\sum _{n=1}^{\infty }n^{n}x^{n}$

(b) $\sum _{n=1}^{\infty }{\frac {(x+1)^{n}}{\sqrt {n}}}$

Solution

Detailed Solution

Return to Sample Exam

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

Difference between revisions of "009C Sample Midterm 2, Problem 4"

Latest revision as of 12:38, 12 November 2017

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