If I construct $$G(x) =\sum_{1}^{\infty} g(x +2n\pi),$$ does this make $G(x)$ $2\pi$-periodic? My understanding is that if $G(x)$ were now $2\pi$-periodic, then that means $G(x) = G(x + 2\pi$) = G(x + $4\pi$), and so on...
does the $2\pi$ and the $4\pi$ just get "absorbed" into the $2n\pi$ in the summation?
Thanks,