For |x| < 1 and any real number alpha: (1 + x)^alpha = 1 + alphax + alpha(alpha-1)$x^2$/2! + alpha(alpha-1)(alpha-2)$x^3$/3! + ...

This is an infinite series (terminating only when alpha is a non-negative integer).

Key special cases: (1+x)^{-1} = 1 - x + $x^2$ - $x^3$ + ... (geometric series) (1+x)^{-2} = 1 - 2x + 3$x^2$ - 4$x^3$ + ... (1+x)^{1/2} = 1 + x/2 - $x^2$/8 + $x^3$/16 - ... (1-x)^{-1/2} = 1 + x/2 + 3$x^2$/8 + 5$x^3$/16 + ...

Application: Approximation. For small x: (1+x)^n ≈ 1 + nx. Example: (1.01)^{10} ≈ 1 + 10(0.01) = 1.1.