How do you find the power series representation for the function #f(x)=1/((1+x)^2)# ?

2 Answers
Sep 7, 2014

By Binomial Series,
#1/{(1+x)^2}=(1+x)^{-2}=sum_{n=0}^{infty}(-1)^n(n+1)x^n#

Let us review the binomial series.
#(1+x)^{alpha}=sum_{n=0}^{infty}C(alpha,n)x^n#,
where #C(alpha,n)# is a binomial coefficient defined by
#C(alpha,n)={alpha(alpha-1)(alpha-2)cdot cdots cdot(alpha-n+1)}/{n!}#

Let us first the binomial coefficients for #f(x)=(1+x)^{-2}#.
Since #alpha=-2#, its binomial coefficient looks like
#C(-2,n)={-2(-2-1)(-2-2)cdot cdots cdot[-2-(n-1)]}/{n!}#
#={-2(-3)(-4)cdots[-(n+1)]}/{n!}#
by factoring out all #-#'s in the numerator,
#={(-1)^n[2cdot3cdot4cdot cdots cdot(n+1)]}/{n!}={(-1)^n(n+1)!}/{n!}#
by dividing the numerator and the denominator by #n!#,
#=(-1)^n(n+1)#

Hence, we have the binomial series
#1/{(1+x)^2}=sum_{n=0}^{infty}(-1)^n(n+1)x^n#.

Mar 11, 2017

#1/(1+x)^2=sum_(n=0)^oo(-1)^(n+1)nx^(n-1)# for #absx<1#

Explanation:

We have the standard power series:

#1/(1-color(blue)x)=sum_(n=0)^oocolor(blue)x^n#

From this we write the power series for #1/(1+x)#:

#1/(1+x)=1/(1-color(blue)((-x)))=sum_(n=0)^oo(color(blue)(-x))^n=sum_(n=0)^oo(-1)^nx^n#

Note that #1/(x+1)=(x+1)^-1#, so #d/dx(1/(1+x))=-(x+1)^-2#:

#d/dx(1/(1+x))=(-1)/(x+1)^2=d/dxsum_(n=0)^oo(-1)^nx^n#

Which can be rewritten as:

#(-1)/(1+x)^2=sum_(n=0)^oo(-1)^nd/dxx^n=sum_(n=0)^oo(-1)^n(nx^(n-1))#

Reversing the signs by multiplying both sides by #-1#:

#1/(1+x)^2=-sum_(n=0)^oo(-1)^n(nx^(n-1))=sum_(n=0)^oo(-1)^(n+1)nx^(n-1)#

Which is convergent for #absx<1#.