Answer
converges
Work Step by Step
Given $$ \sum_{n=3}^{\infty} \frac{3 n+5}{n(n-1)(n-2)}$$
Compare with the convergent series $\displaystyle \sum_{n=3}^{\infty}\frac{1}{n^{2}}$ ($p-$series , p>1) and by using the Limit Comparison Test, we get:
\begin{align*}
\lim_{n\to \infty} \frac{a_n}{b_n}&=\lim_{n\to \infty} \frac{3 n^2(n+5)}{n(n-1)(n-2)}\\
&=3
\end{align*}
Then $\displaystyle \sum_{n=3}^{\infty}\frac{3 n+5}{n(n-1)(n-2)}$ also converges
