TITLE: How to sum this series to infinity: $\sum_{n=0}^{\infty} \frac1{2^{2^n}}$
QUESTION [5 upvotes]: How to sum the series:
$$\sum _{ n=0 }^{ n=\infty  }{ \frac { 1 }{ { 2 }^{ { 2 }^{ n } } }  }$$ 
PS: Just a hint would suffice.

REPLY [2 votes]: By one of the Liouville theorems, this number is transcendental, other similar constructs are $\sum 10^{-n^2}$ and $\sum 10^{-n!}$, or in this context, $\sum 2^{-n^2}$ and $\sum 2^{-n!}$.
So there is no nice formula for this series.