So I have the series -
$$\sum_{n=0}^∞ (-1)^{n+1} \frac{(2x)^n}{n!}$$
I know that by the ratio test, the absolute value of the above series converges for all $x$ (as $\rho < 1$). Since I know that the absolute value of the series converges, can I say that the alternating series also converges or does it not work like that?
If not, I'm not sure what test to use to prove that this alternating series converges for all $x$. Kindly help me understand how I can do that.