Answer
Diverges
Work Step by Step
Given: $f(x)=\lim\limits_{n\to \infty} \cos {n\pi}$
The sequence converges when the limit $\lim\limits_{n\to \infty} a_n$ exists and when the limit $\lim\limits_{n\to \infty} a_n$ does not exist, then sequence diverges.
Here, we have $n=1,2,3,4...$
$\cos \pi=-1; \cos 2\pi=1; \cos 3 \pi =-1;\cos 4\pi =1...$
Thus, we can see that the values for the $\cos$ repeated alternatively. This implies that the value of the function $\cos$ oscillates between $-1$ to $1$.This means that we will not get any definite value.
Hence, $\lim\limits_{n\to \infty} a_n$ does not exist, so the sequence diverges.
