Let X and Y be two independent Poisson random variables, with means λ1 and λ2, respectively. Then, X+Y is a Poisson random variable with mean λ1+λ2. Arguing in a similar way, a Poisson random variable X with parameter t, where t is a positive integer, can be thought of as sum of t independent Poisson random variables X1,X2,…,Xt, each of which has mean 1.

Question

Using the information above, and an appropriate limit theorem, evaluate the following limit:
limn→∞∑k>n+n∞e−nnkk!.