If λ is a sequence K-space and Σ xj is a series in a topological vector space X, the series is said to be λ-multiplier convergent if the series converges in X for every t = {tj} ∈ λ. We show that if λ satisfies a gliding hump condition, called the signed strong gliding hump condition, then the series converge uniformly for t = {tj} belonging to bounded subsets of λ. A similar uniform convergence result is established for a multiplier convergent series version of the Hahn-Schur Theorem.