# Radius of convergence of $\displaystyle \sum_{-\infty}^{\infty}3^{-|n|}z^{2n}, z \in \mathcal{C}$

I want to find out the radius of the following power series of a complex variable:

$\displaystyle \sum_{-\infty}^{\infty} 3^{-|n|} z^{2n}, z \in \mathbb{C}$

The ration test $\displaystyle \lim_{n \to \infty}|\frac{a_m}{a_{m+1}}|$ gives me the radius of convergence as 3 . BUt I am not really sure how to handle the negative power terms. Should I group same co-efficient terms, and complete the square? But then, I dont get a polynomial in a single variable.

Also, how do I evaluate the sum?

Your series is $$1 + \sum_{n=1}^\infty \bigg(\frac{z^2}3\bigg)^n + \sum_{n=1}^\infty \bigg(\frac1{3z^2}\bigg)^n,$$ which converges when both $|z^2/3|$ and $|1/3z^2|$ are less than $1$, that is, when $1/\sqrt3<|z|<\sqrt3$.