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

1 Answer
Sep 13, 2016

Answer:

#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#