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.
The Lagrange remainder is
$$R_n(x) =\frac{f^{(n+1)}(\xi)}{(n+1)!}(x-r)^{n+1} = \frac{(-1)^{n}n!}{(n+1)!\xi^{n+1}}(x-r)^{n+1},$$
where $x, \xi \in (r/2, 3r/2).$
Hence,
$$\left|R_n(x) \right|= \frac{1}{n+1}\frac{|x-r|^{n+1}}{|\xi|^{n+1}} \leqslant \frac{1}{n+1}\frac{|r/2|^{n+1}}{|r/2|^{n+1}} = \frac{1}{n+1}.$$
Thus, the Taylor series converges since $R_n(x) \to 0$ as $n \to \infty$.