Half range sine series: x

For a function on (0, L) the half range sine series is f(x) = Σ bₙ sin(nπx/L), with bₙ = (2/L)∫₀ᴸ f(x) sin(nπx/L) dx. Here L = 2.

Coefficient for f(x) = x:

bₙ = (2/2)∫₀² x sin(nπx/2) dx
   = [ −x cos(nπx/2)·(2/nπ) ]₀² + (2/nπ)∫₀² cos(nπx/2) dx
   = −(4/nπ) cos nπ + (2/nπ)·(2/nπ)[sin(nπx/2)]₀²
   = −4(−1)ⁿ/(nπ) + 0
   = 4(−1)ⁿ⁺¹/(nπ)

Series

x = (4/π) Σ (n=1 to ∞) [(−1)ⁿ⁺¹/n] sin(nπx/2), 0 < x < 2

= (4/π)[ sin(πx/2) − (1/2) sin(πx) + (1/3) sin(3πx/2) − … ]

Check: at x = 1 the series gives (4/π)(1 − 1/3 + 1/5 − …) = (4/π)(π/4) = 1 = x.