Fun with Fourier series

Authors: Canals Gustavo Adolfo

(Submitted on 1 Jun 2008)

Abstract: In this undergraduate-level expository paper, we use computer algebra and graphing capabilities to experiment with elementary Fourier series. We derive some surprising variations on standard formulas for Pi, including Gregory's classic series: Pi/4 = 1 - 1/3 + 1/5 - ... . We find infinite series for which we can multiply the nth term by powers of sin(n)/n while leaving the sum unchanged. For example, for k = 0, 1, 2, and 3, these four sums (written here in Mathematica notation) all equal (Pi - 3)/2: Sum[Sin[3n]/n * (Sin[n]/n)^k, {n, 1, Infinity}].
We also touch on the problem of recovering a function in closed form, given its Fourier series.

adolfocanals@educ.ar
Dr.Física Teórica; PhD-math.
MPI.
adolfocanals@educ.ar