Limsupremum calculation horrible difficulty

[betty2301]

Homework Statement

Calculate the radius of convergence of $\sum_{n=0}^\infty a_{n}z^{2n}$
let $\sum_{n=0}^\infty a_{n}z^{n}$with radius R

Homework Equations

$\limsup|a_n|^{\frac{1}{n}}=\limsup |\frac{a_{n+1}}{a_n}|$

The Attempt at a Solution

$$\limsup|a_n|^{\frac{1}{n}}=\lim_{k\to\infty}|a_{2k}|^{\frac{1}{2k}}$$
then how to write the "ratio"....
the latex is killin me please help....

 

[snipez90]

You could use the limsup formula, but it's easier to think about what the radius of convergence means. Basically you're given that the original series converges for |z| < R and diverges for |z| > R, so what happens when you square z?