Hi, I have some difficulty proving the following
Decide when $\displaystyle \sum _{n=1} ^{\infty} \frac{n^{n}}{a^{n} n!}$ converges when the ratio test fails.
I think that it will diverge, but I'm not sure how to prove it.
Thanks in advance
Hi, I have some difficulty proving the following
Decide when $\displaystyle \sum _{n=1} ^{\infty} \frac{n^{n}}{a^{n} n!}$ converges when the ratio test fails.
I think that it will diverge, but I'm not sure how to prove it.
Thanks in advance
The ratio test fails when $\displaystyle a=\pm e$. At $\displaystyle a=e$, use the comparison test to compare the series with $\displaystyle \sum n^{-1/2}$, using Stirling's formula. At $\displaystyle a=-e$, use the alternating series test.