Prove the Taylor series centered around $\displaystyle 0$ converges to the indicated function.
$\displaystyle \ln(1 + x), x \in [0,1].$
I just have to show the remainder term goes to zero, i.e.,
$\displaystyle f(x) = T_{n}(x) + R_{n}(x).$
So either show $\displaystyle \displaystyle\lim_{n\to\infty}T_{n}(x) = f(x),$
Or, $\displaystyle \displaystyle\lim_{n\to\infty}\frac{f^{(n+1)}(c)}{ (n+1)!}(x - a)^{(n+1)} = 0.$