Fun With Fourier Series

2008/06/01 by Baillie, Robert
#40-01 #42-01 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.0806.0150

Abstract

By using computers to do experimental manipulations on Fourier series, we construct additional series with interesting properties. We construct several series whose sums remain unchanged when the n^th term is multiplied by sin(n)/n. One example is this classic series for π/4: \fracπ4 = 1 - (1)/(3) + (1)/(5) - (1)/(7) + … = 1 ⋅ (sin(1))/(1) - (1)/(3) ⋅ (sin(3))/(3) + (1)/(5) ⋅ (sin(5))/(5) - (1)/(7) ⋅ (sin(7))/(7) + … . Another example is ∑_n=1^∞ (sin(n))/(n) = ∑_n=1^∞ ((sin(n))/(n))² = (π-1)/(2). This paper also discusses an included Mathematica package that makes it easy to calculate and graph the Fourier series of many types of functions.

Related

• Miscellaneous summation, integration, and transformation formulas
• Series expansions for 1/π^m and π^m
• A primer on Fourier Series
• A combinatorial approach to the Fourier expansions of powers of cos and sin
• Sums, series, and products in Diophantine approximation