Proof of divergence/convergence of a series Consider the series :
$$\sum\limits_{n = 1}^\infty  {\frac{{(n + 2)^3 n^\alpha  }}{{\sqrt[3]{{n^2  + 4n + 7\,}}\sqrt {n + 1} }}} $$
where $
\alpha  \in \Re $ . I managed to determine that when $
\alpha  \ge \frac{{ - 13}}{6}$ the series diverges, but what about the other cases ? Can anyone help ?

Hint : Divide denominator and numerator by $n^{(\frac{2}{3}+\frac{1}{2})}=n^\frac{7}{6}$
