--- title: "Fourier series: x − x²" subject: "Mathematics III" chapter: 4 key: "semester_3/mathematics_iii:fourier-series-x-x2" source: https://www.bctengineer.com/q/semester_3/mathematics_iii/fourier-series-x-x2 site: BCT Engineering --- # Fourier series: x − x² Mathematics III · Chapter 4: Fourier Series - Asked in: 2083 Baishakh, 5 marks; 2079 Baishakh, 5 marks; 2075 Asoj, 5 marks; 2072 Kartik, 5 marks; 2067 Asar (asked 5 times) - Video: [Fourier Series of (x-x^2): Full Calculation](https://www.youtube.com/watch?v=yMWKgKi0qDk) — Ayan Sarkar - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/fourier-series-x-x2 - Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-4 ## Question Obtain the Fourier series to represent f(x) = x − x² from x = −π to x = π and hence deduce that π²/12 = 1/1² − 1/2² + 1/3² − 1/4² + ... ## Short answer Fourier series on (-π,π): f(x) = -π^2/3 + Σ_{n=1}^∞ [4(-1)^(n+1)/n^2 cos nx + 2(-1)^(n+1)/n sin nx]. Evaluating at x=0 gives 0 = -π^2/3 + 4(1/1^2 - 1/2^2 +...), hence π^2/12 = 1/1^2 - 1/2^2 + 1/3^2 - ... ## Answer f(x) = x − x² on (−π, π). Use ∫ x² cos nx dx and ∫ x sin nx dx over (−π, π), with cos nπ = (−1)ⁿ. #### Coefficients ```text a₀ = (1/π) ∫_{−π}^{π} (x − x²) dx = (1/π)(−2π³/3) = −2π²/3 aₙ = (1/π) ∫ (x − x²) cos nx dx = −(1/π)∫ x² cos nx dx = −4(−1)ⁿ/n² bₙ = (1/π) ∫ (x − x²) sin nx dx = (1/π)∫ x sin nx dx = 2(−1)ⁿ⁺¹/n ``` (x cos nx and x² sin nx are odd, so they integrate to 0; ∫_{−π}^{π} x² cos nx dx = 4π(−1)ⁿ/n² and ∫_{−π}^{π} x sin nx dx = 2π(−1)ⁿ⁺¹/n.) #### Series ```text x − x² = −π²/3 + Σ_{n=1}^∞ [ 4(−1)ⁿ⁺¹/n² · cos nx + 2(−1)ⁿ⁺¹/n · sin nx ] = −π²/3 + 4cos x − cos 2x + (4/9)cos 3x − ... + 2 sin x − sin 2x + (2/3) sin 3x − ... ``` #### Deduction The function is continuous at x = 0 and f(0) = 0, with sin 0 = 0 and cos 0 = 1: ```text 0 = −π²/3 + 4 [ 1/1² − 1/2² + 1/3² − 1/4² + ... ] ``` **Hence π²/12 = 1/1² − 1/2² + 1/3² − 1/4² + ...** ∎ ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. In x - x^2, which part contributes to the cosine terms? A. x (odd) B. Both x and x^2 C. -x^2 (even) D. Neither Answer: C. Even parts give cosine terms, odd parts sine terms. (Key point: Odd/even parts) 2. a0 = A. 2 pi^2/3 B. -pi^2/3 C. 0 D. -2 pi^2/3 Answer: D. (1/pi) integral of -x^2 = -2pi^2/3, so the constant is a0/2 = -pi^2/3. (Key point: Constant term) 3. bn = A. 2(-1)^(n+1)/n B. 2(-1)^n/n C. 4(-1)^(n+1)/n^2 D. 0 Answer: A. From integral of x sin nx. (Key point: Sine coefficients) 4. Putting x = 0 gives pi^2/12 = 1 - 1/4 + 1/9 - ... because A. 0 = pi^2/3 + 4[...] B. 0 = -pi^2/3 + 4[1 - 1/4 + 1/9 - ...] C. pi^2 = 4[...] D. f(0) = 1 Answer: B. f(0) = 0, cos 0 = 1, sin 0 = 0; the series is continuous there. (Key point: Deduction) ## Close variants set in other exams - [Fourier series of a function](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Afourier-series-of-a-function) — Mathematics III, chapter 4, asked 6× ## Related in Mathematics III - [Fourier series: x − x²](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Afourier-series-x-x2-four) — Mathematics III, chapter 4 - [Fourier series: |x|](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Afourier-series-x) — Mathematics III, chapter 4, asked 6× - [Fourier series: eˣ](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Afourier-series-e) — Mathematics III, chapter 4, asked 6× - [Fourier series: x²](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Afourier-series-x2) — Mathematics III, chapter 4, asked 9× - [Fourier series: x²](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Afourier-series-x2-two) — Mathematics III, chapter 4, asked 2× - [Fourier series: x sin x](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Afourier-series-x-sin-x-two) — Mathematics III, chapter 4, asked 7× ## Related in other subjects - [Line voltage Fourier series: triplen harmonics absent](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_electronics%3Asquare-wave-inverter-triplen-absent) — Power Electronics, chapter 5 - [Fourier series of rising sawtooth −V0 to V0](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Afourier-sawtooth-minus-v0-to-v0) — Electrical Circuit Theory, chapter 6 - [Fourier series of falling sawtooth V to 0](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Afourier-falling-sawtooth-v-to-0) — Electrical Circuit Theory, chapter 6 - [Fourier series of ramp wave 0 to 20](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Afourier-ramp-0-to-20-period) — Electrical Circuit Theory, chapter 6