Define $\displaystyle f(t)=e^{-t} on [-\pi,\pi) $ and extend f to be $\displaystyle 2\pi$-periodic.Find the complex Fourier series of f.

Then, apply Parseval's relation to f to evaluate $\displaystyle /sum {1/(1+k^2)} $.

I know how the complex fourier series looks like but I don't know how to extend it periodically