Question

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

Answer 1

The answer is `=1/8-3/16x+3/16x^2-5/32x^3+o(x^3)`

Explanation:

The binomial series is

`(a+b)^n=a^n+na^(n-1)b+((n)(n-1))/(1*2)a^(n-2)b^2+((n)(n-1)(n-2))/(1*2*3)a^(n-3)b^3+.....`

Here,

`a=2`

`b=x`

and

`n=-3`

Therefore,

`1/(2+x)^3=(2+x)^(-3)`

`=2^(-3)+(-3)*(2^(-4))*x+(-3*-4)/(2)*2^(-5)x^2+(-3*-4*-5)/(6)2^(-6)x^3+....`

`=1/8-3/16x+3/16x^2-5/32x^3+o(x^3)`