Answer
Conditionally Convergent
Work Step by Step
A series $ \Sigma a_n$ is defined as absolutely convergent when the series $ \Sigma |a_n|$ is convergent.
From the series, we notice that
$ \Sigma_{n=1}^{\infty} |\dfrac{(-1)^n}{\sqrt n+\sqrt {n+1}}|= \Sigma_{n=2}^{\infty} \dfrac{1}{n^{1/2}}$
This corresponds to a p-series with common ratio $r=\dfrac{1}{2}$; when the common ratio $r \gt 1$, then a p-series is convergent.
Therefore, the given series is Not Absolutely Convergent.
The Alternating Series Test states:
Consider a series $\Sigma a_n$ such that $p_n=(-1)^n q_n$; $q_n \geq 0$ for all $n$
If the following conditions are satisfied then the series converges:
a) $\lim\limits_{n \to \infty} q_n=0$;
b) $q_n$ is a decreasing sequence.
We have: $B_n=\dfrac{1}{\sqrt n+\sqrt {n+1}}$
a) $\lim\limits_{n \to \infty} p_n=\lim\limits_{n \to \infty}\dfrac{1}{\sqrt n+\sqrt {n+1}}=0$
b) The series is a decreasing sequence.
This implies that the given series is Conditionally Convergent by the Alternating Series Test.
