How do you use the binomial series to expand (1+2x)^(-2)?

Jan 13, 2017

${\left(1 + 2 x\right)}^{- 2} = 1 - 4 x + 12 {x}^{2} - 32 {x}^{3} + \ldots$

Explanation:

The Binomial Series give us the expansion:

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

And so:

 (1+2x)^(-2) = 1 + (-2)(2x) + (-2)(-3)(2x)^2/(2!) +
" " (-2)(-3)(-4)(2x)^3/(3!) + ...
$\text{ } = 1 - 4 x + 6 \frac{4 {x}^{2}}{2} - 24 \frac{8 {x}^{3}}{6} + \ldots$
$\text{ } = 1 - 4 x + 12 {x}^{2} - 32 {x}^{3} + \ldots$