Question

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

Answer 1

`(1-x^3)^(-1/2)=1+1/2x^3+3/8x^6+5/16x^9+35/128x^12+.......`

Explanation:

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

According to binomial series `(1-a)^n=1-na+(n(n-1))/(2!)a^2-(n(n-1)(n-2))/(3!)a^3+(n(n-1)(n-2)(n-3))/(4!)a^4-.......`

Hence `(1-x^3)^(-1/2)=1-(-1/2)x^3+((-1/2)(-1/2-1))/(2!)x^6-((-1/2)(-1/2-1)(-1/2-2))/(3!)x^9+((-1/2)(-1/2-1)(-1/2-2)(-1/2-3))/(4!)x^12-.......`

or `(1-x^3)^(-1/2)=1+1/2x^3+((-1/2)(-3/2))/(2!)x^6-((-1/2)(-3/2)(-5/2))/(3!)x^9+((-1/2)(-3/2)(-5/2)(-7/2))/(4!)x^12+.......`

or `(1-x^3)^(-1/2)=1+1/2x^3+3/8x^6+15/48x^9+105/384x^12+.......`

or `(1-x^3)^(-1/2)=1+1/2x^3+3/8x^6+5/16x^9+35/128x^12+.......`