Closed form of Gaussian-like summation?

Question

I want to evaluate $$\sum^\infty_{n=1}e^{-n^2}$$

My attempt is to evaluate $$\oint_{|z|=\infty}\pi\cot(\pi z)e^{-z^2}dz$$ and prove the circular integral tends to zero.

Unfortunately, this is clearly not the case.

Any suggestions?