--- title: "Fourier series: x²" subject: "Mathematics III" chapter: 4 key: "semester_3/mathematics_iii:fourier-series-x2-two" source: https://www.bctengineer.com/q/semester_3/mathematics_iii/fourier-series-x2-two site: BCT Engineering --- # Fourier series: x² Mathematics III · Chapter 4: Fourier Series - Asked in: 2070 Chaitra, 5 marks; 2069 Chaitra, 5 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-two - Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-4 ## Question Obtain the Fourier series for f(x) = x² in the interval −π < x < π and hence prove that Σ 1/n² = 1/1² + 1/2² + 1/3² + ... = π²/6. ## Short answer x^2 is even, b_n=0. a_0=2pi^2/3, a_n=4(-1)^n/n^2, giving x^2 = pi^2/3 + 4 sum_{n=1}^infinity (-1)^n cos(nx)/n^2. Setting x=pi yields sum_{n=1}^infinity 1/n^2 = pi^2/6. ## Answer f(x) = x² on (−π, π) is even, so bₙ = 0. ```text a₀ = (2/π) ∫₀^π x² dx = 2π²/3 aₙ = (2/π) ∫₀^π x² cos nx dx = 4(−1)ⁿ/n² ``` (by parts: ∫₀^π x² cos nx dx = [x² sin nx/n + 2x cos nx/n² − 2 sin nx/n³]₀^π = 2π(−1)ⁿ/n².) #### Fourier series ```text x² = π²/3 + 4 Σ_{n=1}^∞ (−1)ⁿ cos nx / n² ``` #### Deduction x² is continuous at x = π (f(π) = f(−π) = π²). Put x = π, cos nπ = (−1)ⁿ, so (−1)ⁿ(−1)ⁿ = 1: ```text π² = π²/3 + 4 Σ 1/n² 4 Σ 1/n² = 2π²/3 ``` **Hence Σ 1/n² = 1/1² + 1/2² + 1/3² + ... = π²/6.** ∎ ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. x^2 on (-pi,pi) = A. pi^2/3 + 4 sum (-1)^n cos nx/n^2 B. pi^2/3 + 4 sum cos nx/n^2 C. pi^2/3 - 4 sum (-1)^n cos nx/n^2 D. 2 pi^2/3 + sum cos nx/n^2 Answer: A. From a0, an. (Key point: Series) 2. To get sum 1/n^2, we put A. x = 0 B. x = pi C. x = pi/2 D. x = 2 pi Answer: B. cos n pi = (-1)^n so (-1)^n(-1)^n = 1. (Key point: Point chosen) 3. pi^2 = pi^2/3 + 4 sum 1/n^2 gives sum 1/n^2 = A. pi^2/12 B. pi^2/8 C. pi^2/6 D. pi^2/3 Answer: C. 4 sum = 2 pi^2/3. (Key point: Result) ## Related in Mathematics III - [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: |x|](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Afourier-series-x) — 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: ((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: 2x − x²](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Afourier-series-2x-x2) — 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× ## 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 - [State and prove Fourier convolution theorem](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Aconvolution-theorem-fourier-prove) — Applied Mathematics, chapter 4, asked 3× - [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