Consider the series
$$\sum_{n=0}^{\infty}\cos^n(n)$$
I think that the root test is inconclusive, because
$$\limsup_n \sqrt[n]{|\cos^n(n)|}=\limsup_n|\cos(n)|\leq 1$$
once we can approximate $\pi$ by rational numbers, there will always be some $i$ and $j\in\mathbb{N}$ such that $|j\pi-i|<\varepsilon$, for every $\varepsilon>0$ that we choose. And in this case $|\cos(i)-1|<\delta$.
Nevertheless, it seems that it converges. I can't think of any other convergent series to compare with it.
My question is: how can I prove that this series converges?
Edit: Actually, this series diverges, as you can see in tmyklebu's answer. I made a fortran program and here are some values of the sequence of the partial sums:
n S_n
10 1.5898364866640549
100 7.8365722183614510
1000 24.825953005207236
10000 79.232008037801393