Convergence of the series $\sum\limits_{n = 4}^\infty {\frac{{n + 1}}{{(n + 5)(n + 4)(n - 3)}}}$? To analyze the convergence of the $$\sum\limits_{n = 4}^\infty {\frac{{n + 1}}{{(n + 5)(n + 4)(n - 3)}}}$$ series I used the criterion of integral $$\displaystyle\int_4^\infty {\frac{{x + 1}}{{(x + 5)(x + 4)(x - 3)}}dx},$$ but calculate this improper integral is a very laborious task. Is there a shorter way? What criteria of convergence would be most effective or simple?

Assume that the series begins at $n=4$. Then, we have

$$n+5\ge n$$

$$n+4\ge n$$

$$n-3\ge \frac14 n$$

$$n+1\le 2n$$

Therefore, have

$$\frac{n+1}{(n+5)(n+4)(n-3)}\le \frac{2n}{\frac14 n^3}=8\frac1{n^2}$$

Finally, using the result $\sum_{n=1}^\infty \frac1{n^2}=\frac{\pi^2}{6}$ reveals

$$\sum_{n=4}^{\infty}\frac{n+1}{(n+5)(n+4)(n-3)}\le 8\sum_{n=4}^{\infty}\frac1{n^2}=4\left(\frac{\pi^2}{3}-3\right)$$

and the series converges by the comparison test.