I'm interested in approximating $\sum\_{n=0}^\infty\frac{(-1)^n}{\log^\* (n+2)}$.

Here, $\log^\*$ is the [iterated log function](https://en.wikipedia.org/wiki/Iterated_logarithm).

I tried several solvers (e.g., [Wolfram](https://www.wolframalpha.com/input/?i=sum_0%5Einf%20%28-1%29%5En%2Flog*%28n%2B2%29)), but none seem to yield a solution.
Any ideas?