Problem with limits involving a summation

[straycat]

Hello all,

I am trying to prove that the following is true:

$$lim_{M \rightarrow \infty} \sum_{P = (\frac{1}{N}-\delta)M}^{(\frac{1}{N}+\delta)M}<br /> \frac{(N-1)^{{M-P}}M!}{P!(M-P)!N^{M}} \rightarrow 1$$

where $$P$$, $$M$$, and $$N$$ are integers, and $$\delta$$ is an arbitrarily small positive number (less than $$1/N$$).

Any ideas on how I might approach this?

David

 

[Hurkyl]

Well, since you're doing statistics, can you interpret that summation as a probability? Maybe you know some things about the probability distribution that might help.