A Fourier series is a way to expand a periodic function in terms of sines and cosines. The Fourier series is named after Joseph Fourier, who introduced the series as he solved for a mathematical way to describe how heat transfers in a metal plate.

The GIFs above show the 8-term Fourier series approximations of the square wave and the sawtooth wave.

Mathematica code:

f[t_] := SawtoothWave[t] 

T = 1; 

nmax = 18; 

a0 = (2/T)*Integrate[f[t], {t, -(T/2), T/2}] 

anlist = Table[(2/T)*Integrate[f[t]*Cos[(2*Pi*n*t)/T], 

         {t, -(T/2), T/2}], {n, 1, nmax}] 

bnlist = Table[(2/T)*Integrate[f[t]*Sin[(2*Pi*n*t)/T], 

         {t, -(T/2), T/2}], {n, 1, nmax}] 

fs[t_, nmax_] := a0/2 + Sum[anlist[[n]]*Cos[(2*Pi*n*t)/T] + 

         bnlist[[n]]*Sin[(2*Pi*n*t)/T], {n, 1, nmax}] 

Manipulate[Column[{Plot[{f[t], fs[t, nmax0]}, {t, -1, 1}, 

         PlotRange -> All, AxesLabel -> {"t", "f(t)"}, 

         PlotStyle -> {{Thick, Black}, {Thick, Red}}, 

         ImageSize -> 700, AspectRatio -> 1/2.8], 

         Row[{"f(t)=", fs[t, nmax0]}]}], {nmax0, 1, nmax, 1}]       

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