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, $b_{n}=\frac{lnn}{\sqrt n}$
1. $b_{n}=\frac{lnn}{\sqrt n}$, is a decreasing sequence.
2. $\lim\limits_{n \to \infty}b_{n}=\lim\limits_{n \to \infty}\frac{lnn}{\sqrt n}$
$=\lim\limits_{n \to \infty} \frac{2\sqrt n}{n}$
$=0$
Hence, the series is convergent by AST.
