TITLE: Why does the taylor series of $\ln (1 + x)$ only approximate it for $-1<x \le 1$?
QUESTION [7 upvotes]: I'm looking for an intuitive understanding instead of a formal proof. Thanks for the help.

REPLY [8 votes]: Write $$\log(1+x)= \sum_{n=1}^ \infty (-1)^{n+1}\frac {x^{n}}{n}.$$
Now you can apply the ratio test to get the radius of convergence. Finally, you need to check convergence at the boundary points.