The Fourier series exponential form is
$$\sum_{k=-n}^n c_n e^{2\pi ikx}$$
Is
$e^{-2\pi ik} = 1$ and why
and why is $-e^{-\pi ik}$ equal to ${(-1)^{k+1}}$ and $e^{-\pi ik} = {(-1)^{k}}$, for this I can imagine for $k=0$ that both are equal but for $k>0$ I really don't get it.
And also could you also suggest or list what other primitives might be helpful in computing fourier series, which I was not able to mention here or haven't found out yet. Any help is appreciated.