On a bounded increasing power series

Abstract

is divergent. Given any e> 0 we may by choosing k large enough ensure that (3) an= O(nl+E). However the series (2) is certainly Abel summable and so the stronger condition (4) an O(n-1) would imply the convergence of (2), by Littlewood's Tauberian theorem. Thus Shapiro's example shows that we cannot weaken (4) to a condition of the form (3) in Littlewood's theorem by making the compensating assumption thatf has bounded variation on [0, 1). In this note we prove a stronger negative result than that of Shapiro. THEOREM. Let d(n) be positive for all positive integers n, and let d(n) T oo as n-> oo. Then there is a function f of the form (1) which is increasing and bounded on [0, 1) and for which (2) is divergent although (5) 1 a.j k2 (k_ 1). Put

Locations

• Proceedings of the American Mathematical Society - View - PDF