Answer
Divergent
Work Step by Step
Alternating series test:
Suppose that we have a series $\Sigma a_n$, such that $a_{n}=(-1)^{n}b_n$ or $a_{n}=(-1)^{n+1}b_n$, where $b_n\geq 0$ for all $n$.
Then if the following two condition are satisfied, the series is convergent.
1. $\lim\limits_{n \to \infty}b_{n}=0$
2. $b_{n}$ is a decreasing sequence.
In the given problem, $b_{n}=cos\frac{\pi}{n}$
$\lim\limits_{n \to \infty}b_{n}=\lim\limits_{n \to \infty}cos\frac{\pi}{n}$
$=cos(0)$
$=1$
Thus, the limit is not zero.
Hence, the given series is divergent by the divergence test.
