On classes of everywhere divergent power series

We prove the everywhere divergence of series $$ sum_{n=0}^infty a_n e^{irho_n}e^{inx}, quadtext{and}quad sum_{n=0}^infty {(-1)}^{[rho_n]}a_n cos nx, $$ for sequences a _n and ρ _n satisfying some extremal conditions. These results generalize some well known examples of everywhere divergent power and trigonometric series.