Using the Central Limit Theorem to prove a limit

Use the central limit theorem to prove that, as M tends to infinite:

exp(-2M) * SUM(exp(2Mn)/n!) --> 1/2 . The sum goes from n=0 to n=M.

I don't know how to format it correctly, sorry.

I sort of know what to do with this question. I need to find the right variables so the left side of the CLT boils down to what's above, and the right would be the normal distribution, presumably from 0 to infinite to get a half. A hint or a correction if I'm wrong will be brilliant to start me off, thanks :)

James