Binomial expansion: what does 'range of validity' really mean?

Published 2026-04-22 · Written by Pete Bromfield

You know the binomial expansion for $(1 + x)^n$ where $n$ is any rational number. But every mark scheme wants you to write $|x| < 1$ next to your expansion. Here's why.

The rule

The expansion $(1 + x)^n = 1 + nx + \dfrac{n(n-1)}{2!}x^2 + \dots$ converges to the true value of $(1+x)^n$ only when $|x| < 1$.

Outside that range, the series either diverges (goes to infinity) or converges to a wrong number. Which is why every "expand up to $x^3$" question at A-Level ends with "state the range of validity".

Substitution changes the range

Expand $(1 + 2x)^{1/2}$ up to $x^3$ and state the range of validity.

Here $x$ is replaced by $2x$. So the range of validity is $|2x| < 1$, i.e. $\boxed{|x| < \tfrac{1}{2}}$.

The mark scheme call-out

Range of validity is a B1 — a standalone mark. You get it just for writing $|x| < \tfrac{1}{2}$. Skipping this line loses 1/4 of the question.

Drill 10 binomial-expansion questions on Skills Practice → The Binomial Expansion.