Convergence of a series

[ganondorf29]

1. The problem statement, all variables and given/known data
Find the radius of convergence and the interval of convergence for

\sum_{n=0}^\infty \frac{x^n}{n3^n}





2. Relevant equations



3. The attempt at a solution
Ok, so I first applied the ratio test.

\lim_{n\rightarrow\infty}

|
\frac{x^{n+1}}{(n+1)3^(n+1)} /
\frac{x^n}{n3^n}
|


After some cancellations I got


L = |x/3|




Does this mean that the interval of convergence is from -3<x<3 ?

[Dick]

Yes.

[Office_Shredder]

You have to check the endpoints of your interval separately to see if the series converges there since the ratio test doesn't give any information when the limit is 1