Answer
Divergent
Work Step by Step
Given $$\sum_{n=1}^{\infty} \frac{2}{\sqrt{n}+2}$$
Use the Limit Comparison Test with $a_n =\dfrac{2}{\sqrt{n}+2}$ and $b_n=\dfrac{1}{ \sqrt{n}}$
\begin{align*}
\lim _{n \rightarrow \infty} \frac{a_{n}}{b_{n}}&=\lim _{n \rightarrow \infty} \frac{2\sqrt{n}}{\sqrt{n}+2}\\
&=2
\end{align*}
Since $\displaystyle \sum_{n=1}^{\infty} \frac{1}{ \sqrt{n}}$ is divergent ($p-$ series $p<1$) , then $\displaystyle\sum_{n=1}^{\infty} \frac{2}{\sqrt{n}+2}$ is also divergent.