<s>[INST] 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?
 [/INST] 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.
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