One commonly-derived example of these constructs has a coefficient of four over n pi when n is odd, and zero when even; that one evaluates to a square wave.
Multiplying one of these by itself, integrating from negative L to L and multiplying by one over L gives a useful identity involving the sum of the squares of their coefficients; that identity is used alongside one of these in order to solve the Basel problem. They can be smoothened by applying an expression involving the Lanczos sigma factor; that method is used to eliminate the (*) Gibbs phenomenon, which occurs when they are calculated around discontinuities. One commonly-derived example of these constructs has a coefficient of four over n pi when n is odd, and zero when even; that one evaluates to a square wave. They are generally an infinite sum of sine and cosine functions. For 10 points, name these series, which are decompositions of complex oscillating functions into an infinite series of simpler ones.
Multiplying one of these by itself, integrating from negative L to L and multiplying by one over L gives a useful identity involving the sum of the squares of their coefficients; that identity is used alongside one of these in order to solve the Basel problem.
Fourier series [do NOT accept "Fourier transform", but DO accept Fourier expansion or Fourier decomposition; just the word "Fourier" is acceptable after "series", but should be prompted before]