Applying the Ratio Test In class we were told to apply the ratio test whenever we saw a factorial in the series. My problem asks to find the interval of convergence of 
$$\sum_{k} k!x^k$$
So I began by rewriting the problem as the limit of: $$\frac{(k+1)!x^{k+1}}{k!x^k}$$
Do the $k!$ cancel out?

Let $u_k(x)=k! x^k$ then using the ratio test we get
$$\left|\frac{u_{k+1}(x)}{u_k(x)}\right|=(k+1)|x|\xrightarrow{k\to\infty}\ell(=0)<1\iff x=0$$
hence the radius of convergence is $R=0$ and the series is convergent only for $x=0$.
