Answer
converges
Work Step by Step
Given $$\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{\sqrt{n}+\sqrt{n+1}}$$
This is an alternating series. Consider $ a_{n}=\dfrac{1}{\sqrt{n}+\sqrt{n+1}}$, since $\{a_n \}$ is a decreasing sequence and
\begin{align*}
\lim _{n \rightarrow \infty} a_{n}&=\lim _{n \rightarrow \infty} \dfrac{1}{\sqrt{n}+\sqrt{n+1}}\\
&=0
\end{align*}
Then by using the Leibniz Test, the given series converges.
