Determine whether the following series converges/diverges:
$$\sum_{k = 1}^\infty \frac{k\sin(1+k^3)}{k+\ln k}.$$
The problem is with $\sin(1+k^3)$. It is also negative and so I do not know which test to use. I did not find any similar problem anywhere.
Could anyone give me a hint, please? Have a nice day! Thanks!