Let $r$ be any real number with $0 < r < 1$. Then, of course, there is a closed expression for $$\sum_{k=0}^\infty r^k = \frac{1}{1-r}$$ But does there also exist a closed expression for $$\sum_{k=0}^\infty r^{k^2} \qquad ?$$ Decent approximations will be appreciated as well.
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8That's more-or-less a theta function https://en.wikipedia.org/wiki/Theta_function#Jacobi_theta_function – Angina Seng May 19 '19 at 15:49
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3CAS says: $\frac{1}{2} (\vartheta _3(0,r)+1)$ where: $\vartheta _3(0,r)$ is theta function. – Mariusz Iwaniuk May 19 '19 at 15:52