sum n=o to oo (2n)!(x/2)^n
= (2n+2)!(x/2)^(n+1) / (2n)!(x/2)^n
= (2n+2)(2n+1)(x/2)
where do i go from there? ><
Note,
$\displaystyle (2n)!(x/2)^n = \frac{(2n)!}{2^n}\cdot x^n$
Ratio test for $\displaystyle x\not =0$,
$\displaystyle \left| \frac{(2n+2)!}{2^{n+1}} \cdot \frac{2^n}{(2n)!} \cdot x\right| = \frac{(2n+2)(2n+1)}{2}\cdot |x|\to +\infty$
So, this convergense only for $\displaystyle x=0$