Answer
The series is divergent by the Limit Comparison Test.
Work Step by Step
Use the Limit Comparison Test. $\lim\limits_{n \to \infty}\frac{a_n}{b_n}$
Let $a_n=\frac{1}{2n+1}$ and $b_n=\frac{1}{n}$.
Then $\lim\limits_{n \to \infty}\frac{a_n}{b_n}= \lim\limits_{n\to\infty} \left(\frac{1}{2n+1}\cdot\frac{n}{1}\right)=\lim\limits_{n \to \infty}\frac{n}{2n+1}$
Divide the top and bottom by $n$.
$\lim\limits_{n \to \infty}\frac{1}{2+\frac{1}{n}}\to \frac{1}{2}>0.$
The series $\sum_{n=1}^\infty\frac{1}{n}$ is the Harmonic series which is known to diverge. Thus, the series $\sum_{n=1}^\infty\frac{1}{2n+1}$ is also divergent by the Limit Comparison Test.
