---
title: "Half range sine series: x"
subject: "Mathematics III"
chapter: 4
key: "semester_3/mathematics_iii:half-range-sine-series-x-four"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/half-range-sine-series-x-four
site: BCT Engineering
---

# Half range sine series: x

Mathematics III · Chapter 4: Fourier Series

- Asked in: 2070 Asar, 5 marks
- Video: [Half range sine series/ Fourier series/ PU 2018/ Engineering mathematics 3/ Third semester](https://www.youtube.com/watch?v=s__Bkco7Upg) — Rupak Deo
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/half-range-sine-series-x-four
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-4

## Question

Express f(x) = x as a half range sine series in 0 < x < π.

## Short answer

bₙ = (2/π)∫₀^π x sin nx dx = 2(−1)ⁿ⁺¹/n. The half-range sine series is x = 2 Σ_{n=1}^∞ [(−1)ⁿ⁺¹/n] sin nx.

## Answer

The half range sine series on (0, π) is f(x) = Σ bₙ sin nx, with bₙ = (2/π)∫₀^π f(x) sin nx dx.

```text
bₙ = (2/π)∫₀^π x sin nx dx
   = (2/π)[ −x cos nx/n + sin nx/n² ]₀^π
   = (2/π)(−π cos nπ / n)
   = 2(−1)ⁿ⁺¹/n
```

**Answer:**

x = 2 Σ (n=1 to ∞) [(−1)ⁿ⁺¹/n] sin nx,  0 < x < π

= 2( sin x − sin 2x/2 + sin 3x/3 − sin 4x/4 + … )

Check: at x = π/2 this gives 2(1 − 1/3 + 1/5 − …) = 2(π/4) = π/2.

## Check yourself

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

1. A half-range sine series on (0, pi) represents f(x) with which expression?
   A. Sum of bn cos nx
   B. Sum of bn sin nx
   C. a0/2 + sum an cos nx
   D. a0 only
   Answer: B. Sine series uses only sin nx terms. (Key point: Half-range choice)

2. How is the integral of x sin nx evaluated?
   A. -cos nx/n^2
   B. x cos nx/n
   C. x sin nx/n
   D. By parts: -x cos nx/n + sin nx/n^2
   Answer: D. Parts with u = x gives that antiderivative. (Key point: Integration method)

3. What is bn?
   A. 2/n
   B. 2(-1)^(n+1)/n
   C. (-1)^(n+1)/n
   D. 2(-1)^n/n
   Answer: B. (2/pi)(-pi cos n pi/n) = -2(-1)^n/n. (Key point: Coefficient)

4. At x = pi/2 the series gives 2(1 - 1/3 + 1/5 - ...) which equals?
   A. pi/2
   B. pi/4
   C. pi
   D. 1
   Answer: A. The Leibniz series sums to pi/4, so 2(pi/4) = pi/2. (Key point: Check at pi/2)

## Close variants set in other exams

- [Half range sine series: x²](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ahalf-range-sine-series-x2) — Mathematics III, chapter 4, asked 3×
- [Half range sine series: x](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ahalf-range-sine-series-x-two) — Mathematics III, chapter 4, asked 3×

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