# How do you use the binomial series to expand 1 / (1 +x^4)?

Sep 13, 2016

$\frac{1}{1 + {x}^{4}} = 1 - {x}^{4} + {x}^{8} - {x}^{12} + {x}^{16} - {x}^{20} + {x}^{24.} \ldots \ldots \ldots \ldots \ldots \ldots .$

#### Explanation:

Binomial theorem gives the expansion of ${\left(1 + x\right)}^{n}$ as

(1+x)^n=1+nx+(n(n-1))/(2!)x^2+(n(n-1(n-2)))/(3!)x^3+(n(n-1)(n-2)(n-3))/(4!)x^4+....................

Hence $\frac{1}{1 + {x}^{4}} = {\left(1 + {x}^{4}\right)}^{- 1}$

= 1+(-1)x^4+((-1)(-2))/(2!)x^8+((-1)(-2)(-3))/(3!)x^12+((-1)(-2)(-3)(-4))/(4!)x^16+....................

= $1 - {x}^{4} + {x}^{8} - {x}^{12} + {x}^{16} - {x}^{20} + {x}^{24.} \ldots \ldots \ldots \ldots \ldots \ldots .$

Note that it is convergent only for $x < 1$