Question

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

Answer 1

`f(x) = 1 -1/2 x^3 + 3/8x^6 - 5/16 x^9...`

Explanation:

The binomial series tell us that:

`(1+x)^n = 1+nx + (n(n-1))/(2!)x^2 +(n(n-1)(n-2))/(3!)x^3 + ...`

And so for the given function:

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

Then with `n=-1/2`, and replacing `x` in the definition with `x^3` we have:

`f(x) = 1+(-1/2)(x^3) + ( (-1/2)(-3/2) )/2(x^3)^2 + ( (-1/2)(-3/2)(-5/2))/6 (x^3)^3 + ...`

`" " = 1 -1/2 x^3 + (3/4)/2x^6 - (15/8)/6 x^9 + ...`

`" " = 1 -1/2 x^3 + 3/8x^6 - 5/16 x^9...`