Radius and Interval of Convergence for (3x-2)^2/n 3^n Series

[aydin]

Homework Statement

find the radius of convergence and interval of convergence of the series

∞
Σ (3x-2)^2 / n 3^n
n=1

Homework Equations

The Attempt at a Solution

 

[Edellaine]

You need to clean up your notation a little bit, but the ROC for this series is easily found using any number of tests. Try the ratio test or the limsup test.

 

[HallsofIvy]

$$\sum_{n=1}^\infty \frac{(3x-2)^n}{n n!}$$
is, I think, what you want.

Ratio test is probably best.
$$a_n= \frac{|3x-2|^n}{n n!}$$
and
$$a_{n+1}= \frac{|3x-2|^{n+1}}{(n+1) (n+1)!}$$

so the ratio is
$$\frac{|3x-2|^{n+1}}{(n+1)(n+1)!}\frac{n n!}{|3x-2|^n}= |3x-2|\frac{n}{(n+1)^2}$$
What is the limit of that as n goes to infinity? If that is less than 1, the series will converge.