Binomial Expansion for Rational Index (Pure Mathematics 3 – 9709) — Mathematics

Introduction

In Pure Mathematics 1 you used the binomial theorem for positive integer indices, which always terminates after a finite number of terms. For rational indices (fractions, negative integers, and negative fractions), the expansion becomes an infinite series that only converges — that is, gives a valid, finite sum — when the variable satisfies a strict condition on its size. This is the version of the binomial expansion tested throughout 9709 Paper 3, appearing in questions on approximations, partial fractions, and series work. Mastering it is essential for a wide range of examination contexts.

Core Concept

The generalised binomial expansion states:

(1+x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \cdots

valid for $|x| < 1$ , where $n \in \mathbb{Q}$ (any rational number).

Key observations:

When $n$ is a positive integer, the series terminates (all terms from the $(n+1)$ th onward are zero). For all other rational $n$ , the series is infinite.
The series is only valid (convergent) for $|x| < 1$ , i.e. $-1 < x < 1$ . Outside this interval the partial sums grow without bound and the expansion is meaningless.
The expression must start with 1. If you have $(a + bx)^n$ , you must first factorise to write it as $a^n\!\left(1 + \tfrac{b}{a}x\right)^n$ , then apply the standard formula with $x$ replaced by $\tfrac{b}{a}x$ , adjusting the validity condition accordingly.

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Introduction

Core Concept

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