# Binomial Theorem related proofs

1. Dec 6, 2011

### h.shin

1. The problem statement, all variables and given/known data
Let a be a fixed positive rational number. Choose (and fix) a natural number M>a.
Use (a^n)/(n!)$\leq$(a^M/(M!))(a/M)^(n-M) to show that, given e>0, there exists an N$\in$$N$ such that for all n$\geq$N, (a^n)/n! < e.

2. Relevant equations

3. The attempt at a solution
In a previous problem, I saw that when M>n then (a^n)/(n!)<(a^M/(M!))(a/M)^(n-M). So I thought i could maybe use that to come up with a N in relations to e. but i'm not so sure how to do this. I know the equations are long and ugly, but please help.