Proving Convergence of the Taylor Series for 1/(1-x) as n Approaches Infinity

[freshman2013]

Homework Statement

How do I prove that Rn(x) is zero as n approaches infinity for the nth taylor polynomial of 1/(1-x)
when x is between -1<x<1

Homework Equations

1/(1-x)
Rn(x)=M*(x-a)^(n+1)/(n+1)!
nth derivative of 1/(1-x) is n!/(1-x)^n+1

The Attempt at a Solution

the nth derivative of 1/(1-x) is n!/(1-x)^(n+1) so the max of the n+1 derivative between 0 and d if abs(d)<1 is (n+1)!(d)^(n+1)/(n+1)!(1-d)^(n+2). I simplify it and get d^(n+1)/(1-d)^(n+2) But there seems to be values of d between -1 and 1 such that when n approaches infinity, Rn approaches infinity

 

[mfb]

You have to evaluate your derivatives at x=0 to get the individual terms of the taylor expansion.