The Question:
Find the radius of convergence and the interval of convergence of $\displaystyle\sum\frac{n^2x^n}{2\cdotp 4\cdotp 6 \cdots 2n}$
My Work
$$\left|\frac{a_{n+1}}{a_n}\right| =\frac{(n+1)^2|x|}{n^2(2n+2)} = \frac{n^3(1/n + 2/n^2 1/n^3)|x|}{n^3(2+2/n)}$$
which will approach $0$ as $n\rightarrow \infty$ for all values of $x$ which implies Radius of Convergence $=\infty$ and Interval of Convergence $=(-\infty,\infty)$
My Question
Just looking to see if I made any mistakes.