---
title: "Fourier series: x³"
subject: "Mathematics III"
chapter: 4
key: "semester_3/mathematics_iii:fourier-series-x3"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/fourier-series-x3
site: BCT Engineering
---

# Fourier series: x³

Mathematics III · Chapter 4: Fourier Series

- Asked in: 2075 Chaitra, 5 marks; 2074 Chaitra, 5 marks; 2069 Asar, 5 marks; 2068 Baishakh, 5 marks; 2067 Magh (asked 5 times)
- Video: [Que-3 | Fourier series f(x)=x^3 |x³  B.tech | Engineering mathematics-II | in Hindi | BY SURAJ YADAV](https://www.youtube.com/watch?v=kBopn9LSsjg) — Suraj Academy
- 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-x3
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-4

## Question

Define Fourier series for a function f(x). Obtain the Fourier series for f(x) = x³, −π ≤ x ≤ π.

## Short answer

Fourier series: f(x)=a0/2 + Σ(an cos nx + bn sin nx) with integrals for coefficients. For the odd function f(x)=x^3, a0=an=0, bn = 2(-1)^(n+1)(π^2/n - 6/n^3). The series is x^3 = 2 Σ (-1)^(n+1)(π^2/n - 6/n^3) sin nx for -π<x<π.

## Answer

#### Definition
A function f(x) satisfying the Dirichlet conditions in (−π, π) (single-valued, finitely many maxima/minima and finite discontinuities) has the Fourier series

```text
f(x) = a₀/2 + Σ_{n=1}^∞ (aₙ cos nx + bₙ sin nx)
a₀ = (1/π) ∫_{−π}^{π} f(x) dx
aₙ = (1/π) ∫_{−π}^{π} f(x) cos nx dx
bₙ = (1/π) ∫_{−π}^{π} f(x) sin nx dx
```

#### f(x) = x³ on (−π, π)
x³ is an odd function, so a₀ = 0, aₙ = 0 and

```text
bₙ = (2/π) ∫₀^π x³ sin nx dx
```

Integrate by parts repeatedly: ∫₀^π x³ sin nx dx = −π³(−1)ⁿ/n + 6π(−1)ⁿ/n³. So

```text
bₙ = 2(−1)ⁿ⁺¹ [ π²/n − 6/n³ ]
```

#### Series
```text
x³ = 2 Σ_{n=1}^∞ (−1)ⁿ⁺¹ (π²/n − 6/n³) sin nx
   = 2(π² − 6) sin x − 2(π²/2 − 6/8) sin 2x + 2(π²/3 − 6/27) sin 3x − ...
```

(The series converges to x³ for −π < x < π, and to 0 at x = ±π, the average of the end values.)

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Which is part of the Dirichlet conditions?
   A. f must be even
   B. f must be continuous everywhere
   C. Finitely many maxima/minima and finite discontinuities
   D. f must be zero at ends
   Answer: C. Single-valued, finite extrema and finite jumps. (Key point: Dirichlet)

2. x^3 is odd, so
   A. bn = 0
   B. only cosine terms remain
   C. a0 = pi
   D. a0 = an = 0 and only sine terms remain
   Answer: D. Odd functions have sine series. (Key point: Symmetry)

3. bn =
   A. 2(-1)^(n+1)[pi^2/n - 6/n^3]
   B. 2(-1)^n[pi^2/n - 6/n^3]
   C. 2(-1)^(n+1)[pi^2/n + 6/n^3]
   D. (4/pi)(-1)^(n+1)/n
   Answer: A. From repeated integration by parts. (Key point: bn)

4. At x = pi the series converges to
   A. pi^3
   B. 0, the average of the end values
   C. -pi^3
   D. pi^3/2
   Answer: B. x^3 jumps from pi^3 to -pi^3 in the periodic extension. (Key point: Endpoints)

## Close variants set in other exams

- [Fourier series: eˣ](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Afourier-series-e) — Mathematics III, chapter 4, asked 6×

## Related in Mathematics III

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- [Fourier series: x − x²](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Afourier-series-x-x2-five) — Mathematics III, chapter 4
- [Fourier series: e^(3x)](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Afourier-series-e-3x) — Mathematics III, chapter 4

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