Question

How do you use the binomial series to expand `4*sqrt(1+x)`?

Answer 1

The answer is `=4(1+x/2-x^2/8+x^3/16+....)`

Explanation:

Use `(a+b)^n=a^n+(""_1^n)a^(n-1)b+(""_2^n)a^(n-2)b^2.....`
`=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....`
If we use the binomial development, we get
`4(1+x)^(1/2)=4(1+x/2+((1/2)(_1/2))/(1*2)x^2+((1/2)(-1/2)(-3/2))/(1*2*3)x^3+....)`

`=4(1+x/2-x^2/8+x^3/16+....)`