Answer
Convergent
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 conditions 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, $b_{n}=sin\frac{\pi}{n}$
which satisfies both conditions of the Alternating Series Test as follows:
1. $b_{n}=sin\frac{\pi}{n}$, is decreasing.
Here, $\frac{\pi}{n}$ is decreasing and $sinx$ is increasing on $[0,\pi/2]$
which implies $sin\frac{\pi}{n+1}\lt sin\frac{\pi}{n}$
for $n\gt 2$
2. $\lim\limits_{n \to \infty}b_{n}=\lim\limits_{n \to \infty}sin\frac{\pi}{n}$
$=sin(0)$
$=0$
Hence, the given series is convergent by the Alternating Series Test.
