Tail Probability and Divergent Series

TL;DR

This paper explores the relationship between measure theory, divergent series, and probability, showing that for certain functions and divergent series, specific convergent series can be constructed with bounds related to the integral of the function.

Contribution

It introduces a measure-theoretic approach to connect divergent series with probability and number theory, revealing new inequalities and relationships.

Findings

Existence of sequences making series convergent and bounded by the integral

New inequalities linking expectation and divergent series behavior

Connections between probability theory and number theory

Abstract

From mostly a measure-theoretic consideration, we show that for every nonnegative, finite, and $L^{1}$ function on a given finite measure space there is some nontrivial sequence of real numbers such that the series, obtained from summing over the term-by-term products of the reals and the summands of any divergent series with positive, vanishing summands such as the harmonic series, is convergent and no greater than the integral of the function. In terms of inequalities, the implications add additional information on mathematical expectation and the behavior of divergent series with positive, vanishing summands, and establish in a broad sense some new, unexpected connections between probability theory and, for instance, number theory.