--- title: "Fourier series: x²" subject: "Mathematics III" chapter: 4 key: "semester_3/mathematics_iii:fourier-series-x2" source: https://www.bctengineer.com/q/semester_3/mathematics_iii/fourier-series-x2 site: BCT Engineering --- # Fourier series: x² Mathematics III · Chapter 4: Fourier Series - Asked in: 2082 Shrawan, 4+1 marks; 2081 Chaitra (new course), 4 marks (asked 2 times) - Video: [Find a Fourier Series to represent the function f(x) = X² from x=(-π ,π). //Fourier Series](https://www.youtube.com/watch?v=MdPS3iWu9-Q) — Pathshala Club Jsr - 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-x2 - Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-4 ## Question Find the Fourier series for the function f(x) = x² in the interval −π ≤ x ≤ π and deduce that π²/12 = 1/1² − 1/2² + 1/3² − ... ## Short answer x^2 is even, so b_n=0. a_0 = 2pi^2/3, a_n = 4(-1)^n/n^2. The Fourier series is x^2 = pi^2/3 + 4 sum_{n=1}^infinity (-1)^n cos(nx)/n^2. Setting x=0 gives pi^2/12 = 1/1^2 - 1/2^2 + 1/3^2 - ... ## Answer f(x) = x² on (−π, π) is even, so bₙ = 0. #### Coefficients ```text a₀ = (2/π) ∫₀^π x² dx = 2π²/3 aₙ = (2/π) ∫₀^π x² cos nx dx = (2/π) [ x² sin nx/n + 2x cos nx/n² − 2 sin nx/n³ ]₀^π = (2/π)(2π (−1)ⁿ/n²) = 4(−1)ⁿ/n² ``` #### Series ```text x² = π²/3 + 4 Σ_{n=1}^∞ (−1)ⁿ cos nx / n² = π²/3 − 4cos x + cos 2x − (4/9) cos 3x + ... ``` #### Deduction Put x = 0 (x² is continuous there, value 0): ```text 0 = π²/3 + 4 [ −1/1² + 1/2² − 1/3² + ... ] 4 [ 1/1² − 1/2² + 1/3² − ... ] = π²/3 ``` **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. x^2 is even, so A. an = 0 B. bn = 0 C. a0 = 0 D. series has sin terms Answer: B. Only cosines. (Key point: Symmetry) 2. a0 = A. pi^2/3 B. 4 pi^2/3 C. 2 pi^2/3 D. 2 pi/3 Answer: C. (2/pi)(pi^3/3); constant term a0/2 = pi^2/3. (Key point: a0) 3. an = A. 4/n^2 B. -4(-1)^n/n^2 C. 2(-1)^n/n^2 D. 4(-1)^n/n^2 Answer: D. (2/pi)(2 pi(-1)^n/n^2). (Key point: an) 4. Putting x = 0 yields A. 4[1 - 1/4 + 1/9 - ...] = pi^2/3 B. 4[1 + 1/4 + 1/9 + ...] = pi^2/3 C. 4[...] = pi^2 D. [...] = pi^2/3 Answer: A. 0 = pi^2/3 - 4[...]; hence pi^2/12. (Key point: Deduction) ## Close variants set in other exams - [Fourier series: ((π − x)/2)²](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Afourier-series-x-2-2) — Mathematics III, chapter 4, asked 5× - [Fourier series: (π − x)²/4](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Afourier-series-x-2-4) — Mathematics III, chapter 4, asked 5× - [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: 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× - [Fourier series: 2x − x²](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Afourier-series-2x-x2) — Mathematics III, chapter 4, asked 5× ## Related in Mathematics III - [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× - [Fourier series: x − x²](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Afourier-series-x-x2) — Mathematics III, chapter 4, asked 6× - [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: x³](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Afourier-series-x3) — 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: |sin x|](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Afourier-series-sin-x) — Mathematics III, chapter 4, asked 2× ## Related in other subjects - [Fourier transform vs Fourier series](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Afourier-transform-vs-series) — Applied Mathematics, chapter 4 - [Derive exponential Fourier series (CT)](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Actfs-exponential-derivation) — Signals and Systems, chapter 2, asked 15× - [Derive the discrete-time Fourier series](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adtfs-derivation) — Signals and Systems, chapter 2, asked 15× - [Derive trigonometric Fourier series (CT)](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Actfs-trigonometric-derivation) — Signals and Systems, chapter 2, asked 3×