Binomial Expansion for General (Rational) n

When n is not a positive integer — for example n = 1/2, n = -1, or n = -3/2 — we can still expand (1 + x)^n, but the expansion is an infinite series and is only valid for |x| < 1. This extends the binomial theorem and is tested at A-Level (9MA0).

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The General Binomial Expansion

For any rational number n and |x| < 1:

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

This is an infinite series (it never terminates unless n is a non-negative integer).

Key condition: the expansion is valid only when |x| < 1.

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How This Differs from Positive Integer n

Feature	Positive integer n	General/rational n
Number of terms	Finite (n + 1 terms)	Infinite
Values of x	All real x	Only
Coefficient formula	n!/(r!(n-r)!)	n(n-1)(n-2)...(n-r+1)/r!
Result	Exact	Approximation (using first few terms)

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Expanding (1 + x)^n for Rational n

Worked Example 1