Let $r>0$. I want to use the Lagrange Form of the Remainder (as stated, say, here) to prove that the Taylor series of $\log(x)$ centered at $r$ converges to $\log$ on $(\frac{r}{2}, \frac{3r}{2})$.
I've calculated the Taylor series to get $\sum_{n=0}^{\infty}\frac{(-1)^{n}(n-1)!}{(n!)r^n}(x-r)^n$ but am lost on how to do this problem. I'd appreciate some help.