Math Help - Question on Alternating Series

1. Question on Alternating Series

I have a question on interpreting sigma notation. Is this accurate?

$e^{-x^2} = \sum_{n=0}^\infty \frac{(-x^2)^n}{n!} = \sum_{n=0}^\infty \frac{x^{2n}}{n!}$

It seem like it must be, but I'm not sure, so I wanted to confirm. It seems to me that if you have a negative variable raised to an even power, it must always be positive. But then, this series is no longer alternating, which seems wrong.

$e^{-x^2} = \sum_{n=0}^\infty \frac{(-x^2)^n}{n!} = \sum_{n=0}^\infty \frac{x^{2n}}{n!}$
It should be $e^{-x^2} = \sum_{n=0}^\infty \frac{(-x^2)^n}{n!} = \sum_{n=0}^\infty \frac{(-1)^nx^{2n}}{n!}$