Question

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

Answer 1

`(1+2x)^(-2) = 1-4x+12x^2-32x^3 + ...`

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!) + ...`
`" "= 1-4x+6(4x^2)/2-24(8x^3)/6 + ...`
`" "= 1-4x+12x^2-32x^3 + ...`