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 $\displaystyle \sum_{n=1}^{\infty}a_n$ converges, then $\displaystyle \lim_{n\to\infty}a_n = 0$.
Looking at the sum $\displaystyle \sum_{n=1}^{\infty}\frac{a^n}{n!}$, use the ratio test. $\displaystyle \lim_{n\to\infty}\frac{\frac{a^{n+1}}{(n+1)!}}{\fr ac{a^n}{n!}} = \lim_{n\to\infty}\frac{a}{n+1} = 0$
Thus you sum converges, and by the theorem $\displaystyle \lim_{n\to\infty}\frac{a^n}{n!}=0$