# convergence of an infinite series

Decide when $\sum _{n=1} ^{\infty} \frac{n^{n}}{a^{n} n!}$ converges when the ratio test fails.
The ratio test fails when $a=\pm e$. At $a=e$, use the comparison test to compare the series with $\sum n^{-1/2}$, using Stirling's formula. At $a=-e$, use the alternating series test.