# Thread: Proving a taylor series converges.

1. ## Proving a taylor series converges.

Prove the Taylor series centered around $\displaystyle 0$ converges to the indicated function.

$\displaystyle \ln(1 + x), x \in [0,1].$

2. Originally Posted by cgiulz
Prove the Taylor series centered around $\displaystyle 0$ converges to the indicated function.

$\displaystyle \ln(1 + x), x \in [0,1].$
Define "prove". Depending on the class setting your in this could be as simple as considering $\displaystyle \int\frac{dx}{1+x}$ or as difficult as discussing uniform convergence.

3. 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, $\displaystyle \displaystyle\lim_{n\to\infty}\frac{f^{(n+1)}(c)}{ (n+1)!}(x - a)^{(n+1)} = 0.$

4. Here is what I've got so far:

$\displaystyle \ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - ... + (-1)^{n-1}\frac{x^n}{n} + \frac{f^{(n+1)}(c)x^{(n+1)}}{(n+1)!}.$

Now.. can I use L'Hospitals?