Question

How do you use the binomial series to expand `(1+2x)^-3`?

Answer 1

`suma_nx^(n-1)`, through n = 1, 2, 3.., with `a_1=1 and a_n=-2((n+2)/n)a_n-1`, n=2, 3, 4,..., for `|x| < 1/2`.

Explanation:

Use `(1+kx)^(-m)=1-m(kx)/1!+m(m+1)(kx)^2/(2!)+...+(-1)^(n-1)(m(m+1)(m+2)...(m+n-1))(kx)^n/(n!)+...`, for `|kx| < 1`.
`=sum a_nx^(n-1)`, where`a_1=1 and a_n=-k((m+n-1)/n)a_(n-1)`

Here, k=2 and m=3.