sequences and series - What is limit of $sum limits_{n=0}^{infty}frac{1}{(2n)!}$?

What is the limit of the series $1 \over (2n)!$ for n in $[0, \infty)$ ?

$$\sum_{n = 0}^{\infty}{1 \over (2n)!}$$
I've ground out the sum of the 1st 1000 terms to 1000 digits using Python,

(see here ), but how would a mathematician calculate the limit? And what is it?

No, this isn't homework. I'm 73. Just curious.

Thanks

Answer

It's half the sum of $e^1=\sum 1/n!$ and $e^{-1}=\sum (-1)^{n}/n!$ (or $\cosh 1$, in other words).