Here's another series with a similar weird property: a1 + a2 + a3 + ... = a1^2 + a2^2 + a3^2 + ... Specifically: Sum[ Sin[n]/n , {n, 1, Infinity}] = Sum[ (Sin[n]/n)^2 , {n, 1, Infinity}] = (Pi - 1)/2 The Fourier series for this function: f[x_] := -(Pi + x)/2 /; -Pi < x < -1; f[x_] := x(Pi - 1)/2 /; -1 < x < 1; f[x_] := (Pi - x)/2 /; 1 < x < Pi; is: Sum[ Sin[n]/(n^2) * Sin[n x] , {n, 1, Infinity}]. At x = 1, you get the sum of (Sin[n]/n)^2 The function g[x_] := (Pi-x)/2 has the Fourier series Sum[ Sin[n x]/n , {n, 1, Infinity}]. At x = 1, you get the sum of Sin[n]/n. This was problem 6241 in the Monthly about 25 years ago. ----- Schroeppel, Richard wrote:
Fascinating!
I had a different objective in mind when I wrote the problem. My goal was to reach the fairly difficult pi^2/6 result using series manipulation, starting from a definition of pi based on 1 - 1/3 + 1/5 - ... or the like. Ideally, one would avoid calculus and complex integrations and the theory of dilogarithms, or Euler's trick with the power series for sin x.
Rich