$n$ is an integer greater than $1$.
2026-05-06 04:12:45.1778040765
Solution/approximation for $\sum_{z\in\mathbb{Z}} e^{-\frac{(zn+1)^2}{2}}$
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Jacobi theta function.
Definition: $$ \vartheta_3\left( z,q \right) =\sum _{k=-\infty}^{\infty }{q}^{ {k}^{2}}e^{i2kz},\qquad |q|<1, z \in \mathbb C . $$ So we get $$ S_n = \sum_{k=-\infty}^\infty e^{-{(kn+1)^2}/{2}} =e^{-1/2}\vartheta_3\left(\frac{in}{2},e^{-n^2/2}\right) $$