Series $\sum_{n=1}^{\infty} \frac{n^2 - 5n}{n^3 + n + 1}$

Simply use the fact that$$\lim_{n\to\infty}\frac{\dfrac{n^2-5n}{n^3+n+1}}{\dfrac1n}=\lim_{n\to\infty}\frac{1-\dfrac5n}{1+\dfrac1{n^2}+\dfrac1{n^3}}=1$$and that the harmonic series diverges.