real analysis - Does $sum_{n=2}^inftyfrac{coslnln n}{ln n}$ converge?

$$\sum_{n=2}^\infty\frac{\cos\ln\ln n}{\ln n}$$
My idea is
$$-\frac1{\ln n}\le\frac{\cos\ln\ln n}{\ln n}\le\frac1{\ln n}$$
But I don't know if $\sum\frac1{\ln n}$ converges.