General Binomial Expansion - Subtleties (AQA A-Level Mathematics): Flashcards

Valid when $|x|$ is sufficiently small for convergence

Powers increase to infinitely large numbers (diverges)

Higher powers become smaller (converges)

$|ax| < 1$

$|x| < \frac{1}{8}$

Factor out to get $(1+...)^n$ form

$3^{-2}(1 + \frac{2}{3}x)^{-2}$

$|x| < \frac{3}{2}$

$|x| < \frac{1}{7}$

$1 + ax + bx^2 + cx^3 + dx^4 + \ldots$