Convergence/divergence of the series $\sum_{n=4}^\infty\frac{(-1)^n}{\log(\log(n))}$?

We have the following sum:

$$\displaystyle \sum_{n=4}^\infty \dfrac{(-1)^n}{\log(\log(n))}$$

I have a hunch this series is conditionally convergent, but I get nowhere using the ratio test. What test would be best to apply to this particular series?

• I changed $x\to n$ to have the question body agree with the title. – Winther Dec 9 '15 at 0:02

You may just use the alternating series test:

• the function $x \mapsto \dfrac{1}{\log(\log(x))}$ is decreasing,

• as $x \to \infty$, you have $\dfrac{1}{\log(\log(x))} \to 0.$