By Taylor's theorem, $$\ln(x+1) = \sum_{j=1}^{n} (-1)^{j+1}\frac{x^j} n + \frac{(-1)^n}{n+1}(1+c)^{-(n+1)}x^{n+1},$$ for some $c\in (0,x)$. It's easy to see that the second term tends to $0$ as $n$ increases when $x\in [0,1]$. It's also easy to see that it diverges to infinity when $x>1$. For $x\in (-1,0)$, it's not clear whether it gets small or not. Trying out some values, it seems like the error does tend to $0$, but I don't see how to prove this. Taylor's theorem doesn't seem to help here as the second term could be arbitrarily big for some $c$'s in $(0,x)$.
Why does (if it indeed does) this approximation work for $x\in (-1,0)$ and can I get some hint on how to prove this?
Note: I use the notation $(a,b)$ with $a>b$ to mean $(b,a) = \{x: b<x<a\}$.
