calculus - Prove the series $sum_{n=1}^{infty}frac{(-1)^n}{ln{n}+sin{n}}$ converge

How to prove this serie

$$\sum_{n=1}^{\infty}\frac{(-1)^n}{\ln{n}+\sin{n}}$$

converge? I can't do a comparison test with the Leibniz formula for $\pi$ because the series are not $>0$ for all $n$. I can't do a ratio test because I can't compute the limit, the alternating series test can't be applied, the absolute serie is not convergent. I'm out of ideas.