Combining Taylor expansions How do you taylor expand the function $F(x)={x\over \ln(x+1)}$ using standard results? (I know that WA offers the answer, but I want to know how to get it myself.) I know that $\ln(x+1)=x-{x^2\over 2}+{x^3\over 3}+…$ But I don't know how to take the reciprocal. In general, given a function $g(x)$ with a known Taylor series, how might I find $(g(x))^n$, for some $n\in \mathbb Q$?
Also, how might I evaluate expressions like $\ln(1+g(x))$ where I know the Taylor expansion of $g(x)$ (and $\ln x$). How do I combine them?
Thank you.

You can do this by long division:
$\dfrac{x}{x-{x^2\over 2}+{x^3\over 3}-\cdots} = 1 + \dfrac{{x^2\over 2}-{x^3\over 3}+\cdots}{x-{x^2\over 2}+{x^3\over 3}\cdots} = 1 + \frac{x}{2} +  \dfrac{-{x^3\over 12}+\cdots}{x-{x^2\over 2}+\cdots} = 1 + \frac{x}{2} -  \frac{x^2}{12} + \cdots  $
