Answer
Convergent
Work Step by Step
Consider $a_n=\dfrac{n+1}{5n^2(n+3)}$ and $b_n=\dfrac{1}{ n^2}$
Now, $\lim\limits_{n \to \infty}\dfrac{a_n}{b_n} =\lim\limits_{n \to \infty}\dfrac{\dfrac{n+1}{5n^2(n+3)}}{\dfrac{1}{ n^2}}$
Thus, we have $ =\lim\limits_{n \to \infty} \dfrac{1+\dfrac{1}{n}}{5(1+\dfrac{3}{n})}$
or, $\lim\limits_{n \to \infty} \dfrac{1+\dfrac{1}{n}}{5(1+\dfrac{3}{n})}=\lim\limits_{n \to \infty} \dfrac{1+0}{5(1+0)}=\dfrac{1}{5}$
Here, $\Sigma_{n=1}^\infty \dfrac{1}{n^{2}}$ is a convergent series due to the p-series test with $p=2 \gt 1$
Thus the series converges by the limit comparison test.
