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

Oct 17, 2016

Answer:

${\left(1 + {x}^{6}\right)}^{\frac{1}{2}} = 1 + \frac{1}{2} {x}^{6} - \frac{1}{8} {x}^{12} + \frac{1}{16} {x}^{18} - \frac{5}{128} {x}^{24} + \ldots \ldots .$

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 ${\left(1 + {x}^{6}\right)}^{\frac{1}{2}} = 1 + \frac{1}{2} {x}^{6} - \frac{1}{8} {x}^{12} + \frac{1}{16} {x}^{18} - \frac{5}{128} {x}^{24} + \ldots \ldots .$