Question

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

Answer 1

the binomial expansion is `1-4x+12x^2-32x^3+80x^4-192x^5+....`

Explanation:

We must write in the form`(1+b)^n`
The development of `(1+b)^n=1+n/1(b)+(n(n-1))/(1*2)b^2*......`
n can be positive or negative

So we apply this and we get

`1/(1+2x)^2=(1+2x)^(-2)=1+(-2)/(1)(2x)+(-2*-3)/(1*2)(2x)^2+(-2*-3*-4)/(1*2*3)(2x)^3+(-2*-3*-4*-5)/(1*2*3*4)(2x)^4+(-2*-3*-4*-5*-6)/(1*2*3*4*5)(2x)^5+...`
`=1-4x+12x^2-32x^3+80x^4-192x^5+.......`