TITLE: Sum of Infinite Series $1 + 1/2 + 1/4 + 1/16 + \cdots$
QUESTION [5 upvotes]: Everyone knows about the classic 
$$ \sum_{i=1}^{\infty} \dfrac{1}{2^i} = 1 $$
However, is there any way to find
$$ \sum_{i=0}^{\infty} \dfrac{1}{2^{2^i}} = \dfrac12 + \dfrac14 + \dfrac{1}{16} + \dfrac{1}{256} + \cdots $$

REPLY [4 votes]: The sum mentioned by the OP, and sums like it, are in general very difficult to deal with. The OP's series is actually called the Fredholm number, and is transcendental (see the Wikipedia page for Transcendental number). 
Note: This constant is also known as Kempner's number (not to be confused with the Kempner series!).