Evaluate $\sum_{n=0}^\infty \left(\frac12\right)^{n^2}$

Question

Can we evaluate this? $$\sum_{n=0}^\infty \left(\frac12\right)^{n^2}$$ I came up into this while I was thinking of some nested radical problem like: $$\sqrt{7\sqrt{\sqrt{\sqrt{7\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{7\dots}}}}}}}}}$$

Answer 1

If you are familiar with Jacobi Elliptic functions, you will find that $$K = \sum_{n=0}^\infty \left(\frac12\right)^{n^2} = \frac12\left( 1+\vartheta_3(0;\frac12)\right) \approx 1.56446841361 $$ There is no simpler expression for $K$.

Because the series for $K$ approaches rational numbers "quickly" it is likely that it is transcendental. For example, the Fredholm number $$\sum_{n=0}^\infty \left(\frac12\right)^{2^n}$$ has beeen proven to be transcendental.