One of the first examples of Fourier series that a student encounters is something like this: f(x) = x/2, for -Pi < x < Pi. The Fourier series is Sum[ (-1)^(n+1)/n * Sin[n x] ]. A second, more complicated example is: f(x) = -Pi/2, for -Pi < x < 0, f(x) = Pi/2, for 0 < x < Pi. This Fourier series is Sum[ (1 - (-1)^n)/n * Sin[n x] ]. Can one work backwards, from the coefficients to the function? That is: given an expression in n, say, c(n) = (-1)^(n+1)/n, can one find the function that has c(n) as its coefficients? This seems difficult, especially because in the second example, the function consists of two distinct pieces over [-Pi, Pi], and as part of the calculation, you'd have to find the point (x = 0) where the function jumps. One could plot 1000 terms of Sum[ c(n) * Sin[n x] ] for -Pi < x < Pi, then try to guess what the expressions and points of discontinuity of f(x) are. Or, one could compute 1000 terms for many x's and do curve fitting, but guessing would be involved here, too. But is there a systematic method to do this? Bob Baillie