Binomial Expansion: Advanced

This lesson covers the advanced binomial expansion as required by the AQA A-Level Mathematics specification (7357). At AS-Level, you studied the binomial expansion of (a + b)ⁿ for positive integer n. At A-Level, you must also be able to expand (1 + x)ⁿ for rational (including negative and fractional) values of n, determine the range of validity, and use expansions for approximations.

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Recap: Binomial Expansion for Positive Integer n

For a positive integer n:

(a + b)ⁿ = ∑ᵣ₌₀ⁿ ⁿCᵣ aⁿ⁻ʳ bʳ

where ⁿCᵣ = n!/(r!(n − r)!).

This is a finite expansion with (n + 1) terms.

Example: (1 + x)⁴ = 1 + 4x + 6x² + 4x³ + x⁴.

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The General Binomial Expansion for Rational n

When n is not a positive integer (i.e., n is negative, fractional, or zero), the expansion of (1 + x)ⁿ is an infinite series:

(1 + x)ⁿ = 1 + nx + n(n−1)/2! x² + n(n−1)(n−2)/3! x³ + ...

This can be written as:

(1 + x)ⁿ = 1 + nx + n(n−1)x²/2! + n(n−1)(n−2)x³/3! + n(n−1)(n−2)(n−3)x⁴/4! + ...

Validity Condition