I'm having trouble understanding a couple of parts of the proof presented here in Knapp's Basic Real Analysis : Book link from Google (pgs 389-390).

He defines c_n=\frac{1}{2L}\int_{-L}^{L}f(t)e^{-2\pi int/L}dt. He then takes the Fourier series for L=1 on F(x)=\sum_{n=-\infty}^{\infty}f(x+n) and gets F(x)=\sum_{n=-\infty}^{\infty}e^{2\pi i nx}\int_{0}^{1}F(t)e^{-2\pi i nt}dt which seemingly identifies c_n=\int_{0}^{1}F(t)e^{-2\pi i nt}dt. This means to me that F(t)e^{-2\pi int} is an even function of t, but this is not mentioned; am I missing something here?

My second question is when he seemingly substitutes t+k\rightarrow t in the integral \int_0^1 f(t+k)e^{-2\pi int}dt=\int_{k}^{k+1}f(t)e^{-2\pi int}dt. However shouldn't the exponential in the integrand be e^{-2\pi in(t-k)}?

Thank you for any help!