# Evaluating Summations

I want to calculate the infinite sum: $$\sum^{\infty}_{k=1} \frac{e^{-5}5^{2k-1}}{(2k-1)!}$$ I know this series converges by the ratio test. So I must compute the limit: $$\lim_{n \to \infty} \sum^{n}_{k=1} \frac{e^{-5}5^{2k-1}}{(2k-1)!}$$.

Now I can't spot any links with this summation, how would I evaluate it?

-

$$\sinh{x} = \frac12 \left (e^x-e^{-x} \right ) = \sum_{k=1}^{\infty} \frac{x^{2 k-1}}{(2 k-1)!}$$
ah nice.+1 ${}{}{}$ –  Sabyasachi Mar 16 '14 at 13:39