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

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Answer 1 · 2016-09-13T16:49:40

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

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 $x<1$

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