How do you find the power series representation for the function #f(x)=1/((1+x)^2)# ?
2 Answers
By Binomial Series,
Let us review the binomial series.
where
Let us first the binomial coefficients for
Since
by factoring out all
by dividing the numerator and the denominator by
Hence, we have the binomial series
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-color(blue)((-x)))=sum_(n=0)^oo(color(blue)(-x))^n=sum_(n=0)^oo(-1)^nx^n#
Note that
#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+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