TITLE: Detecting a negative coefficient in a power series
QUESTION [15 upvotes]: Suppose that I have an analytic function $f(z)=\sum_{n=0}^\infty a_n z^n$ which converges on some disk around the origin.  
For a particular function I encountered, I wished to prove that every coefficient, $a_n$, is non-negative.  

I am wondering what complex analytic methods exist to detect negative coefficients if all my coefficients are real.  What nice ways are there to check if all of the coefficients of my power series are the same sign?

In more generality, are there methods which detect whether eventually all of the coefficients are of the same sign?  (That is, whether or not there exists $N$ such that for all $n,m>N$, $a_n$ and $a_m$ will be the same sign)
I am really interested in any, and many, thoughts on this problem.  What strategies could possibly work?
Thanks!

REPLY [3 votes]: See slides 37-40 in this talk by Manuel Kauers here 
Also, about some series with integer coefficients, a selection from the dissertation of Ken Ono here
The typical situation for showing that a number is integrally represented by a positive integral quadratic form is to express the theta series as a difference of some natural series, then show that the result is positive for large enough $n,$ as in your question. A seminal result is in Duke and Schulze-Pillot, Representations of integers by positive ternary quadratic forms and equidistribution of lattice points on ellipsoids, Invent. Math. 99 (1990)