Define the $\vartheta :\mathbb{R}^+ \to \mathbb{R}$ by
$$\vartheta (s) = \sum_{m=-\infty}^{\infty} e^{-\pi m^2s}$$
Is this a smooth ($C^{\infty}$) function? I would like to think so, but I'm not sure due to the sum from $-\infty$ to $+\infty$.
On top of this, I was told that I should try to show
$$s^{-1/2}\vartheta(1/s) = \vartheta(s), \quad s > 0$$
To me this smells like I should use Fourier series, but I'm not sure how to use Fourier series on a function with an infinite series.
Could anyone possibly help me show these two facts?