Sum of series $\sum_1 \frac{\cos(x)^{n-1}}{n!}$

Question

I'm asked to find when this series converges and the sum of this series:

$\sum_{n=1}^{\infty} \frac{\cos(x)^{n-1}}{n!}$

I found out, with the ratio criteria that it converges everywhere but when $x=0+k\pi$ but I have no idea how to find the sum of this series.

Answer 1

If $x=2k\pi $ the series becomes $$\sum \frac {1}{n!} $$ which converges by ratio test and the sum is $e $

If $x=(2k+1)\pi$, it becomes $\sum \frac {(-1)^{n-1}}{n!} $ which converges by alternate criteria and the sum is $e^{-1} $.

Answer 2

$\mid \cos x \mid \le 1,$ so the series converges absolutely everywhere on $\mathbb R$ by comparison with the series $\sum{\frac{1}{n!}}=e.$