Homework Help: Argumenting for this inequality involving the residual of a taylor series of ln

1. Oct 7, 2012

lo2

1. The problem statement, all variables and given/known data

OK I have to argument for the fact that this inequality is true, where x > 1.

$|R_n \ln{x}| \leq \frac{1}{n+1}(x-1)^{n+1}$

And I have found out that the residual is equal to this:

$R_n \ln{x} = \frac{1}{n!} \int^x_a{f^{n+1}(t)(x-t)^{n}dt}$

2. Relevant equations

3. The attempt at a solution

So since I have to just argument, not prove.

Can I just look at the parts $\frac{1}{n!}$ and $(x-1)^{n+1}$, and then say that the first one goes towards zero very fastly as n increases whereas the other one goes towards infinity quite rapidly as well. And therefore this inequality must be true.

Also I perhaps should add that this part:

$\frac{(x-1)^{n+1}}{n+1}$

Goes toward infinity as the exponential function out-sprints the linear function. Of course only for x > 2.