You can use Fourier series to prove Weyl equidistribution theorems. Take any irrational number $a$ and look at the fractional parts of $a,2a,3a,...$. Then this sequence is equidistributed in $[0,1]$. This is a special case of the ergodic theorem and is fairly straight forward to prove. Unless you have seen ergodic theory before it's a pretty darn surprising application of Fourier series. See for example Stein and Shakarchi's Fourier Analysis book for a reference.
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