ratio test is inconclusive

Question

I have a series

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

Using ratio test, I got

$={\sum_{k=1}^{\infty} \frac{cos(k+1)}{k*(2*(k+1)^2-1)}}/{\frac{cos(k)}{k*(2*(k^2)-1)}} = \frac{cos(\infty+1)}{cos(\infty)}*\frac{2-(1/(\infty)^2)}{2+4/(\infty)^2)+1/(\infty)^2}=1$ (inconclusive)

But testing it in online calculators yields convergence results. What am I doing wrong?

Answer 1

Directly with comparison:

$$\left|\frac{\cos k}{2k^3-k}\right|\le\frac1{k^3}\implies\;\text{since}\;\;\sum_{k=1}^\infty\frac1{k^3}$$

converges then so our series converges absolutely, and thus it converges.