# Math Help - Real Analysis NEED HELP QUICK

1. ## Real Analysis NEED HELP QUICK

need help with this problem

show that:

lim as n goes to infinity of a^n/n!=0 for all a in R

2. Originally Posted by economist
need help with this problem

show that:

lim as n goes to infinity of a^n/n!=0 for all a in R
There is a theorem that states that if $\sum_{n=1}^{\infty}a_n$ converges, then $\lim_{n\to\infty}a_n = 0$.

Looking at the sum $\sum_{n=1}^{\infty}\frac{a^n}{n!}$, use the ratio test. $\lim_{n\to\infty}\frac{\frac{a^{n+1}}{(n+1)!}}{\fr ac{a^n}{n!}} = \lim_{n\to\infty}\frac{a}{n+1} = 0$

Thus you sum converges, and by the theorem $\lim_{n\to\infty}\frac{a^n}{n!}=0$