TITLE: A function with positive decreasing Taylor coefficients?
QUESTION [7 upvotes]: The function $\frac{x}{\ln(1-x)}$ has a Taylor series $-1+c_1x+c_2x^2+\cdots$ and I want to show $c_1>c_2>\cdots>0$. More generally, is there a result about how a function has positive Taylor coefficients or has decreasing Taylor coefficients?

REPLY [5 votes]: We have
$$-\int_{-1}^0 {1\over {(z+1)^t }}dt={ {-z}\over{\log(1+z)}}, \ \ \ \int_{-2}^{-1}{1\over {(z+1)^t }}dt={{z(z+1)}\over{\log(1+z)}}, \ \ z>0$$
So the k-derivative of these functions is positive when k is even and negative  when k is odd.
So the coefficients of the Taylor series of ${z\over{\log(1-z)}},{{z^2-z}\over{\log(1-z)}}$ are positive, as desired.