Question

What is the binomial expansion of `(2 + 3x)^-2`?

Answer 1

`(2+3x)^(-2) = 1/4(1 - 3x + 27/4x^2 + ...)` when `|x| < 2/3`

Explanation:

When `|x| < 1`
`(1+x)^n = 1 + n/(1!)x + (n(n-1))/(2!)x^2 + (n(n-1)(n-2))/(3!)x^3 + ...`

`(2+3x)^(-2) = 1/4(1+3/2x)^(-2)`
`= 1/4(1 - 2 \times 3/2x + 3 \times (3/2x)^2 + ...)` when `|x| < 2/3`
`= 1/4(1 - 3x + 27/4x^2 + ...)` when `|x| < 2/3`