Unsure if $\sum_{n=1}^{\infty} \frac{1}{(5 + \cos n)^n}$ converges or diverges

I am unsure if the series $$\sum_{n=1}^{\infty} \frac{1}{(5 + \cos n)^n}$$ converges or diverges.

I know that $5+\cos n$ lies between $5-1^n=4$ and $5+1^n=6$, meaning that we can bound the expression as $\frac{1}{4^n} \geq \frac{1}{(3 + \cos n)^n} \geq \frac{1}{6^n}$.

Taking $\sum_{n=1}^{\infty} \frac{1}{4^n}=\frac{1}{3}$ and $\sum_{n=1}^{\infty} \frac{1}{4^n}=\frac{1}{5}$, we get the bound as $\frac{1}{3}\geq \sum_{n=1}^{\infty} \frac{1}{(3 + \cos n)^n} \geq \frac{1}{6}$. But this still does not prove convergence since $\frac{1}{3}$ does not equal $\frac{1}{6}$?

If that is the case and the sequence diverges, how should I best show that the sequence diverges in a formal proof?