Answer
The limit does not exist or diverges.
Work Step by Step
The value of the function $\cos x$ always oscillates in between $-1$ and $+1$. Thus, its acquires no limit and $\cos \pi n=1$ for even values of $n$ and $\cos \pi n=-1$; for odd values of $n$.
Therefore, $\lim\limits_{n \to \infty} \cos \pi n$= Limit does not exist
Hence, the series is divergent.
