009C Sample Midterm 2, Problem 4

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

a) $\sum _{n=0}^{\infty }n^{n}x^{n}$

b) $\sum _{n=0}^{\infty }{\frac {(x+1)^{n}}{\sqrt {n}}}$

Foundations:
Root Test
Ratio Test

Solution:

(a)

Step 1:
We begin by applying the Root Test.
We have
${\begin{array}{rcl}\displaystyle {\lim _{n\rightarrow \infty }{\sqrt {|a_{n}|}}}&=&\displaystyle {\lim _{n\rightarrow \infty }{\sqrt {|n^{n}x^{n}|}}}\\&&\\&=&\displaystyle {\lim _{n\rightarrow \infty }|n^{n}x^{n}|^{\frac {1}{n}}}\\&&\\&=&\displaystyle {\lim _{n\rightarrow \infty }|nx|}\\&&\\&=&\displaystyle {n|x|}\\&&\\&=&\displaystyle {|x|\lim _{n\rightarrow \infty }n}\\&&\\&=&\displaystyle {\infty }\end{array}}$

Step 2:
This means that as long as $x\neq 0,$ this series diverges.
Hence, the radius of convergence is $R=0$ and
the interval of convergence is $\{0\}.$

(b)

Step 1:

Step 2:

Final Answer:
(a)     The radius of convergence is $R=0$ and the interval of convergence is $\{0\}.$
(b)     The radius of convergence is $R=1$ and the interval fo convergence is $(2,4].$

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