Find the radius of convergence of R of the following power series.
$\displaystyle \sum_{k=0}^{\infty}\frac{k2^k}{3^k}x^{2k}$
I don't know what to do with the $\displaystyle x^{2k}$ term.
Thanks
Use the root test,
$\displaystyle \left| \frac{k2^k}{3^k}x^{2k} \right|^{1/k} = \frac{2}{3}k^{1/k}x^2$ the limit of $\displaystyle k^{1/k}\to 1$ thus:
$\displaystyle |a_kx^{2k}|^{1/k} \to \frac{2}{3}x^2$.
We require that,
$\displaystyle \frac{2}{3}x^2 < 1 \implies x^2 < \frac{3}{2}$.