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.

Note that the proposed solution by Andrew assumes $a\ge0$. If $a$ can be negative, then absolute values are needed, as evident by other proposed solutions. In Andrew's solution, $x_{n+1}=x_n\frac{a}{n+1}\Rightarrow \lvert x_{n+1}\rvert=\lvert x_n\rvert\lvert \frac{a}{n+1}\rvert$. Then by the same steps this leads to $\lim \lvert x_n\rvert=0$ and consequently, $\lim x_n=0$.
