I was trying to prove that the Taylor polynomial for ln(1+x) converges to the function for |x|<1 from first principles, and was having some difficulty at the end.
The version of Taylor's Theorem I'm using is the following. Given some function which is n times differentiable in some interval containing a and x+a, then
So, after the usual juggling which you've all seen before, I end up with the correct series, and
I would like to show that this goes to zero only for values of x in the range -1<x<1, but I can't see how to do it. Of course this leads me to think I may have made a mistake somewhere along the line, my understanding of Taylor's theorem is not absolute yet.
Thanks in advance