Half range cosine series: eˣ

Half-range cosine series on (0, 1) (l = 1): f(x) = a₀/2 + Σ aₙ cos nπx, with a₀ = 2∫₀¹ f dx and aₙ = 2∫₀¹ f cos nπx dx.

Coefficients

a₀ = 2 ∫₀¹ eˣ dx = 2(e − 1)

Use ∫ eˣ cos nπx dx = eˣ (cos nπx + nπ sin nπx)/(1 + n²π²):

aₙ = 2 [ eˣ (cos nπx + nπ sin nπx)/(1 + n²π²) ]₀¹
   = 2 [ e(−1)ⁿ − 1 ]/(1 + n²π²)

Answer:

eˣ = (e − 1) + 2 Σ_{n=1}^∞ { [ (−1)ⁿ e − 1 ] / (1 + n²π²) } cos nπx,   0 < x < 1

First terms: a₁ = −2(e + 1)/(1 + π²) ≈ −0.68, a₂ = 2(e − 1)/(1 + 4π²) ≈ 0.085, so eˣ ≈ 1.718 − 0.68 cos πx + 0.085 cos 2πx − ...