The Fourier series is a branch of Fourier analysis that aims to decompose a periodic function into a sum of exponentials (or trigonometric functions) with different frequencies and magnitudes, in this particular demonstration, we are defining $f(t)$ to be a periodic complex function with $t\in[0, 1]$
Represent $f(t)$ as a sum of exponential functions rotating at frequencies of $0, 1, -1, 2, -2, ...$ rotations per $t$: