Infinite limit of factorial series

[flyerpower]

Homework Statement

The Attempt at a Solution

I have no idea where to start from.
I know that e = 1/0! + 1/1! + 1/2! + 1/3! + ... + 1/n! + ...
and tried to separate the factorial off the geometric series but had no success.

 

[tarantella]

We start from $$\frac{2^{k-1}(k-1)}{(k+1)!} =\frac{2^{k-1}(k+1-2)}{(k+1)!} =<br /> \frac{2^{k-1}}{k!}-\frac{2^k}{(k+1)!}$$ and we recognize a telescopic series.