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Fourier series: |x|

Short answer

|x| is even, so bₙ=0. a₀=π, and aₙ=0 for even n, -4/(πn²) for odd n. The Fourier series is |x| = π/2 - (4/π)(cos x + cos 3x/3² + cos 5x/5² + ...), -π≤x≤π.

Find the Fourier series representation of f(x) = |x| in [−π, π].

f(x) = |x| on (−π, π) is an even function, so bₙ = 0.

Coefficients

a₀ = (2/π) ∫₀^π x dx = (2/π)(π²/2) = π
aₙ = (2/π) ∫₀^π x cos nx dx
   = (2/π) [ x sin nx/n + cos nx/n² ]₀^π
   = (2/(πn²)) [ (−1)ⁿ − 1 ]

So aₙ = 0 for even n, and aₙ = −4/(πn²) for odd n.

Series

|x| = π/2 − (4/π) [ cos x/1² + cos 3x/3² + cos 5x/5² + ... ]

✓ Answer checked by Claude Opus 5.5 (high effort)

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