Originally Posted by

**Stonehambey** Hi,

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

$\displaystyle f(x+a) = f(a) + \sum_{k=1}^{n-1}f^{(k)}(a)\frac{x^k}{k!} + f^{(n)}(a + \theta x)\frac{x^n}{n!}$

for some $\displaystyle \theta \in (0,1)$

So, after the usual juggling which you've all seen before, I end up with the correct series, and

$\displaystyle \left|R_n(f) \right|= \left| \frac{x^n f^{(n)}(1+\theta x)}{n!} \right| = \left| \frac{x^n}{n(1+\theta x)^n} \right|$

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.