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4.1PureStretch

Expanding (1 + x)ⁿ

In Year 1 you expanded (1+x)n(1+x)^n for positive integer nn. Now nn can be any rational number — negative or fractional. The expansion becomes an infinite series that is only valid when ∣x∣<1|x| < 1.

30 min Binomial Expansion
Edexcel A level Maths: 4.1 Binomial ExpansionCurated by Kepler, not produced by us. Watch it before the notes below.
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What you'll be able to do

  • Use the general binomial expansion for any rational n
  • Find the first few terms of an infinite expansion
  • State the range of validity |x| < 1
  • Recognise when the series converges
1

The general formula

For any rational nn and ∣x∣<1|x| < 1, the expansion continues forever. Each term builds from the last by multiplying by a falling factor and dividing by the next integer.

(1+x)n=1+nx+n(n−1)2!x2+n(n−1)(n−2)3!x3+⋯(1+x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \cdots
Valid for |x| < 1 when n is not a positive integer.
2

Validity

When nn is a positive integer the expansion is finite\textbf{finite} and valid for all xx. Otherwise it is an infinite\textbf{infinite} series, valid only for ∣x∣<1|x| < 1 — outside this range the terms grow and the series diverges.

Tip — Always state the range of validity: |x| < 1 for (1 + x)ⁿ.

3

Worked example

Find the first four terms of (1+x)−2(1+x)^{-2}.

Example 1Expand (1+x)−2(1+x)^{-2} up to the term in x3x^3.
1n=−2n=-2: 1+(−2)x+(−2)(−3)2!x2+(−2)(−3)(−4)3!x31 + (-2)x + \frac{(-2)(-3)}{2!}x^2 + \frac{(-2)(-3)(-4)}{3!}x^3.
2=1−2x+3x2−4x3+⋯= 1 - 2x + 3x^2 - 4x^3 + \cdots
Answer
1−2x+3x2−4x3+⋯1 - 2x + 3x^2 - 4x^3 + \cdots, valid for ∣x∣<1|x| < 1.

Formula recap

(1+x)n=1+nx+n(n−1)2!x2+⋯(1+x)^n = 1 + nx + \tfrac{n(n-1)}{2!}x^2 + \cdots
General binomial, any rational n.
∣x∣<1|x| < 1
Range of validity.

Common mistakes to avoid

Forgetting to state |x| < 1.
The expansion is only valid for |x| < 1 unless n is a positive integer.
Using factorials of n when n is negative or fractional.
Use the product form n(n−1)(n−2)… in the numerator; ⁿCᵣ is undefined for non-integer n.

Key takeaways

  • For any rational n, (1+x)ⁿ = 1 + nx + n(n−1)/2! x² + …
  • Infinite series, valid only for |x| < 1 (unless n is a positive integer).
  • Build each term from the previous falling-factor pattern.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

  1. The expansion of (1+x)n(1+x)^n for non-integer nn is valid when:

    • ∣x∣<1|x| < 1
    • ∣x∣>1|x| > 1
    • all xx
    • x>0x > 0
    Show answer and explanation

    Answer: ∣x∣<1|x| < 1

    Infinite series converges only for |x| < 1.

  2. The coefficient of x2x^2 in (1+x)n(1+x)^n is:

    • n(n−1)2\frac{n(n-1)}{2}
    • n2n^2
    • n2\frac{n}{2}
    • n(n−1)n(n-1)
    Show answer and explanation

    Answer: n(n−1)2\frac{n(n-1)}{2}

    n(n−1)/2!.

  3. The term in xx of (1+x)−2(1+x)^{-2} is:

    • −2x-2x
    • 2x2x
    • −x-x
    • 4x4x
    Show answer and explanation

    Answer: −2x-2x

    nx = −2x.

  4. The x2x^2 term of (1+x)−2(1+x)^{-2} is:

    • 3x23x^2
    • −3x2-3x^2
    • 2x22x^2
    • 6x26x^2
    Show answer and explanation

    Answer: 3x23x^2

    (−2)(−3)/2 = 3.

  5. When nn is a positive integer, the expansion is:

    • finite, valid for all x
    • infinite, |x|<1
    • never valid
    • valid for x>1
    Show answer and explanation

    Answer: finite, valid for all x

    Terminates after n+1 terms.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

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