Function series of the form $$f(x) = \sum\limits_{n = 0}^N {c_n f_n (x)} $$ are considered under the constraintf(x)≥0 in a given intervala≤x≤b. The cone in teh spaceR N+1 of the coefficientsc n which is determined by the positivity constraint is approximated numerically by a polyhedral cone. A numerical estimate for the error involved is given and it is shown how it may be reduced. A special series of Jacobi polynomials is discussed and new estimates for the range of parameters for which this series is non-negative are obtained.