Question

How do you use the binomial series to expand `(1+x^6)^(1/2)`?

Answer 1

`(1+x^6)^(1/2)=1+1/2x^6-1/8x^12+1/16x^18-5/128x^24+.......`

Explanation:

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^6)^(1/2)=1+(1/2)x^6+((1/2)(1/2-1))/(2!)x^12+((1/2)(1/2-1)(1/2-2))/(3!)x^18+((1/2)(1/2-1)(1/2-2)(1/2-3))/(4!)x^24+.......`

or `(1+x^6)^(1/2)=1+1/2x^6+((1/2)(-1/2))/(2!)x^12+((1/2)(-1/2)(-3/2))/(3!)x^18+((1/2)(-1/2)(-3/2)(-5/2))/(4!)x^24+.......`

or `(1+x^6)^(1/2)=1+1/2x^6-1/8x^12+1/16x^18-5/128x^24+.......`