real analysis - Does the series $sum_{n=4}^infty frac{(-1)^n}{log log n}$ converge?

Does the series $\sum_{n=4}^\infty \frac{(-1)^n}{\log \log n}$ converge ?

I thought about alternating test, but for some reason this seems to easy. Why does it start with $n=4$? And how do I prove that $\frac1{\log \log n}$ is decreasing ?