Question

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

Answer 1

`1/(1+x^4)=1-x^4+x^8-x^12+x^16-x^20+x^24....................`

Explanation:

Binomial theorem gives the expansion of `(1+x)^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 `1/(1+x^4)=(1+x^4)^(-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....................`

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