TITLE: What is $\sum_{n=1}^{\infty}\frac{1}{2^{n^2}}$ equal to?
QUESTION [3 upvotes]: So we know that 
$$
\displaystyle\sum_{n=1}^{\infty} \frac{1}{2^n} = 1
$$
Is there any information on what 
$$
\displaystyle\sum_{n=1}^{\infty} \frac{1}{2^{n^2}}
$$
equals?

REPLY [0 votes]: For what it's worth, this number expressible in terms of one of the theta funcitons.
$$
\vartheta_3(0,1/2)=1+2\sum_{n=1}^\infty\frac{1}{2^{n^2}}\text{.}
$$
So your number is $$\frac{\vartheta_3(0,1/2)-1}{2}\text{.}$$
{"set_name": "stack_exchange", "score": 3, "question_id": 1586115}