4.1PureStretch

Expanding (1 + x)ⁿ

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

30 min Video by Zeeshan Zamurred Binomial Expansion

Edexcel A level Maths: 4.1 Binomial ExpansionWatch the full walkthrough 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

The general formula

For any rational $n$ and $∣ x ∣ < 1$ , 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 + n x + \frac{n ( n - 1 )}{2 !} x^{2} + \frac{n ( n - 1 ) ( n - 2 )}{3 !} x^{3} + \dots

Valid for |x| < 1 when n is not a positive integer.

Validity

When $n$ is a positive integer the expansion is $finite$ and valid for all $x$ . Otherwise it is an $infinite$ series, valid only for $∣ x ∣ < 1$ — outside this range the terms grow and the series diverges.

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

Worked example

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

n = - 2

1 + (- 2) x + \frac{( - 2 ) ( - 3 )}{2 !} x^{2} + \frac{( - 2 ) ( - 3 ) ( - 4 )}{3 !} x^{3}

= 1 - 2 x + 3 x^{2} - 4 x^{3} + \dots

Answer

1 - 2 x + 3 x^{2} - 4 x^{3} + \dots

, valid for

∣ x ∣ < 1

Formula recap

(1 + x)^{n} = 1 + n x + \frac{n ( n - 1 )}{2 !} x^{2} + \dots

General binomial, any rational n.

∣ 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

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