# Convergence of $\sum_{n=1}^{\infty} \frac{a^{n+k} n !}{n^{n+k}}$ [closed]

Does the series $$\sum_{n=1}^{\infty} \frac{a^{n+k} n !}{n^{n+k}}$$ with $$a>0$$ and $$k \in \mathbb{N}$$

converges?

Using the ratio test already found that it converges for ae, but couldn't find another method for a=e, as the limit in the ratio test goes to 1.

## closed as off-topic by Martin R, José Carlos Santos, blub, mrtaurho, ShaileshApr 17 at 2:48

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$$\frac{a^{n+k}n!}{n^{n+k}}\sim \left(\frac{a}{n}\right)^{n+k}\sqrt{2\pi n}\left(\frac{n}{e}\right)^n=u_n$$
$$u_n = \left(\frac{a}{e}\right)^n \sqrt{2\pi}a^k n^{1/2-k}$$
As you remarked, if $$a>e$$ this series diverges, and converges if $$a. If $$a=e$$, you therefore have
$$u_n = C n^{1/2-k}$$ with $$C$$ a constant, which converges only if $$k-1/2>1$$, that is, $$k>3/2$$.