On Convergence of Partial Sums of Franklin Series to

In this paper, we prove that if { n k } is an arbitrary increasing sequence of natural numbers such that the ratio n k +1 / n k is bounded, then the n k -th partial sum of a series by Franklin system cannot converge to +∞ on a set of positive measure. Also, we prove that if the ratio n k +1 / n k is unbounded, then there exists a series by Franklin system, the n k -th partial sum of which converges to +∞ almost everywhere on [0, 1].