HMMT 2006 U40

Compute

$\sum_{k=1}^{\infty} \frac{3 k+1}{2 k^{3}+k^{2}} \cdot(-1)^{k+1}$

Dataset text NuminaMath olympiads. Quoted with source.

Solutions

No solution is on olymmath yet. Official and human solutions come first when they arrive. AI solutions come after them, with a label.