A question on Power Series Radius Of convergence-need to check answer What is the radius of convergence for the following power series I did some and want to check the answers
$$ \sum\limits_{n=1}^\infty {n^2\over 2^n}x^{n^2} $$
I solved it the following way but I am not sure 
$ a_n = \dfrac{n}{2^{n^1/2}} $ if   $ n = k^2 $ for some natural number $k$, $ 0 $ Otherwise
Thus $\limsup \sqrt[n]{a_n} = \lim\limits_{n\to\infty}\left( \dfrac{\sqrt[n]{n}}{\sqrt{2}}\right) = \dfrac{1}{\sqrt{2}}$
Radius of convergence $=\dfrac{1}{\sqrt{2}}$

For convergence we need $C=\lim \sup |{n^2\over 2^n}x^{n^2}|^{1\over n}<1$  
