Detecting a negative coefficient in a power series

Solution 1:

If all coefficients are positive, then the dominant singularity lies on the positive real axis.

This would give a criterion to weed out some non-positive series. Of course, it is useless for a proof of positivity.

Solution 2:

If your coefficients are integers, an easy way is to find a combinatorial interpretation. This problem, and problems related to it, appear to be hard to answer algorithmically in general: it is not even known how to detect algorithmically if a coefficient of a rational function is ever zero. See this blog post by Terence Tao on the Skolem-Mahler-Lech theorem.