Prove that $\sum _{n=1}^{\infty } \frac{(-1)^{n+n!}}{n}$ converges or diverges

Question

Determine whether the following series $\sum\limits _{n=1}^{\infty } \cfrac{(-1)^{n+n!}}{n}$ converges or diverges.

I tried to use the ratio test: $$ \lim\limits_{n\to\infty} \left|\cfrac{\cfrac{(-1)^{n+1+(n+1)!}}{n+1}{}}{\cfrac{(-1)^{n+n!}}{n}}\right| = 1 $$ But it tells me nothing.

I cannot use the alternating series test:

Let $$a_n = \cfrac{(-1)^{n!}}{n}$$ then $$\exists n \in \mathbb{N} : a_n < 0$$ Choose $n=1$, all other terms are positive.

Using Wolfram Mathematica code:

SumConvergence[(-1)^(n + n!)/n, n]

I got that the series diverges. How should I prove it? Which convergence test should I use?

Answer 1

Hint:

Both $\;n\;$ and $\;n+n!\;$ have the same parity, for $\;n\ge 2\;$