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