Assume $F(x)$ satisfies

$$\int_{-\infty}^{\infty}|F(x)| d x<\infty$$

and that the series

$$g(\tau)=\sum_{n=-\infty}^{\infty} F(2 n \pi+\tau)$$

converges uniformly in $[0 \leqslant \tau \leqslant 2 \pi]$.

If $\tilde{F}$ is the Fourier transform of $F$, prove that

$$g(\tau)=\frac{1}{2 \pi} \sum_{n=-\infty}^{\infty} \tilde{F}(n) e^{i n \tau}$$

[Hint: prove that $g$ is periodic and express its Fourier expansion coefficients in terms of $\tilde{F}]$.

In the case that $F(x)=e^{-|x|}$, evaluate the sum

$$\sum_{n=-\infty}^{\infty} \frac{1}{1+n^{2}} .$$