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

# Half range cosine series: x

Mathematics III · Chapter 4: Fourier Series

- Asked in: 2067 Asar
- Video: [How To Find The Half Range Cosine Series of f(x)=4x](https://www.youtube.com/watch?v=fMAiHSfr-ik) — DrAvitech
- 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-cosine-series-x-two
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-4

## Question

Express f(x) = x as a cosine half range series in 0 < x < 2.

## Short answer

The half-range cosine series on (0,2) is x = 1 − (8/π²) Σ_{n odd} (1/n²) cos(nπx/2). a₀=2, aₙ=4((−1)ⁿ−1)/(n²π²) giving −8/(n²π²) for odd n and 0 for even n.

## Answer

The half range cosine series on (0, L) is f(x) = a₀/2 + Σ aₙ cos(nπx/L), where a₀ = (2/L)∫₀ᴸ f dx and aₙ = (2/L)∫₀ᴸ f cos(nπx/L) dx. Here L = 2 (the function is extended as an even function).

**Coefficients** for f(x) = x:

```text
a₀ = (2/2)∫₀² x dx = [x²/2]₀² = 2

aₙ = ∫₀² x cos(nπx/2) dx
   = [ x·(2/nπ) sin(nπx/2) ]₀² − (2/nπ)∫₀² sin(nπx/2) dx
   = 0 + (2/nπ)(2/nπ)[cos(nπx/2)]₀²
   = (4/n²π²)(cos nπ − 1)
   = 4((−1)ⁿ − 1)/(n²π²)
```

- n even: aₙ = 0
- n odd: aₙ = −8/(n²π²)

**Series**

x = 1 − (8/π²) Σ (n odd) cos(nπx/2) / n²,  0 < x < 2

= 1 − (8/π²)[ cos(πx/2) + cos(3πx/2)/3² + cos(5πx/2)/5² + … ]

Check: at x = 0 the series gives 1 − (8/π²)(1 + 1/9 + 1/25 + …) = 1 − (8/π²)(π²/8) = 0.

## Check yourself

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

1. The cosine half-range series on (0, 2) uses:
   A. the even extension of f
   B. the odd extension
   C. no extension
   D. a constant extension
   Answer: A. Cosine series are even. (Key point: Extension)

2. a0 for f = x on (0, 2) is:
   A. 1
   B. 2
   C. 4
   D. π
   Answer: B. (2/2)∫0^2 x dx = 2; a0/2 = 1. (Key point: a0)

3. an is:
   A. −4/(nπ) for all n
   B. 8/(n^2π^2) for odd n
   C. −8/(n^2π^2) for odd n, 0 for even n
   D. 0 for all n
   Answer: C. 4((−1)^n − 1)/(n^2π^2). (Key point: an)

4. At x = 0 the series gives 0 using:
   A. Σ1/n^2 = π^2/6
   B. Σ1/n = ln 2
   C. Σ(−1)^n/n = −ln 2
   D. Σ over odd n of 1/n^2 = π^2/8
   Answer: D. 1 − (8/π^2)(π^2/8) = 0. (Key point: Check)

## Close variants set in other exams

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

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## Related in other subjects

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