Is it possible to visualize the graph of a cosine or sine fourier series of an arbitrary function without calculating the coefficients?

Question

Given an arbitrary function (usually a combination of even and odd functions), is it possible to visualize how the sine or cosine fourier series will appear without calculating the coefficients and numerically producing a graph?

Answer 1

As you note in the comments, if your Fourier series is defined on a symmetric interval $[-\ell, \ell]$, then the cosine series of $f$ is the even part of $f$ and the sine series is the odd part of $f$. These parts are given by simple algebraic formulas: $$ f_{\text{even}}(x) = \tfrac{1}{2}\bigl(f(x) + f(-x)\bigr),\qquad f_{\text{odd}}(x) = \tfrac{1}{2}\bigl(f(x) - f(-x)\bigr). $$