Prove of convergence of a serie

Question

Prove if that serie is convergence:

$\sum \limits_{k=1}^\infty (\frac{k^3}{-1-k^4})$
$<=>\sum \limits_{k=1}^\infty -(\frac{k^3}{k^4+1})$
My problem is that I don't know how to do that I tried some different methodes but it doesn't work.Does anyone have an idea?Can I prove it by the p-test although there is a (-1)?

Answer 1

The sum in your question diverges. You can use the limit comparison test with the harmonic series to show that it converges iff the harmonic series converges.

Another way: $$\sum_{k=1}^{\infty}\frac{1}{2k}\leq\sum_{k=1}^{\infty}\frac{1}{k+\frac{1}{k}}=\sum_{k=1}^{\infty}\frac{k^3}{k^4+1}$$

This follows from the fact that $\forall k\in Z^+[2k\geq k+\frac{1}{k}]$