Answer
divergent
Work Step by Step
Given
$$ \sum_{n=1}^{\infty} \frac{ 1}{2n+3}$$
Use the Limit Comparison Test with $a_{n}=\dfrac{ 1}{2n+3}$ and $b_{n}=\dfrac{1}{ n }$
\begin{align*}
\lim _{n \rightarrow \infty} \frac{a_{n}}{b_{n}}&=\lim _{n \rightarrow \infty} \frac{n}{2n+3}\\
&=\frac{1}{2}
\end{align*}
since $\displaystyle\sum_{n=1}^{\infty} \frac{1}{ n }$ is divergent $(p-\text { series } p=1),$ then $\displaystyle\sum_{n=1}^{\infty} \frac{ 1}{2n+3}$ also divergent