Half range sine series: eˣ

The half range sine series on (0, π) is f(x) = Σ bₙ sin nx, with bₙ = (2/π)∫₀^π f(x) sin nx dx.

Coefficient for f(x) = eˣ. Use ∫ eᵃˣ sin nx dx = eᵃˣ(a sin nx − n cos nx)/(a² + n²) with a = 1:

bₙ = (2/π)∫₀^π eˣ sin nx dx
   = (2/π)[ eˣ(sin nx − n cos nx)/(1 + n²) ]₀^π
   = (2/π)·[ −n(−1)ⁿ e^π + n ]/(1 + n²)
   = (2/π)·n[1 − (−1)ⁿ e^π]/(1 + n²)

Series

eˣ = (2/π) Σ (n=1 to ∞) { n[1 − (−1)ⁿ e^π] / (n² + 1) } sin nx, 0 < x < π

= (2/π)[ (1 + e^π)/2 · sin x + 2(1 − e^π)/5 · sin 2x + 3(1 + e^π)/10 · sin 3x + … ]

(The first terms follow by putting n = 1, 2, 3.) A numerical check at x = 1 with many terms gives 2.718, matching e¹.