Does the sequence $a^n/n!$ converge? The sequence when plotted converges to zero because a factorial grows faster than the numerator, but I can not prove that this sequence actually converges.

METHODOLOGY $1$:
Here is another way forward that is very efficient.  
We know from the $n$'th Term Test that if a series converges, then its terms must approach zero.
Inasmuch as the series 
$$\sum_{n=0}^\infty \frac{a^n}{n!}=e^a$$
converges, then the $n$'th Term Test guarantees that $\lim_{n\to \infty}\frac{a^n}{n!}=0$.  And we are done!

METHODOLOGY $2$:
It is easy to show that $n!>(n/2)^{n/2}$.  Therefore, we have
$$\left|\frac{a^n}{n!}\right|\le \left(\frac{\sqrt 2 |a|}{\sqrt n}\right)^n\to 0\,\,\text{as}\,\,n\to \infty$$
