Power series determined by an experiment on the unit interval

TL;DR

This paper studies the expected minimal index at which linear combinations of a fixed nonnegative sequence and a random sequence from the unit interval exceed a threshold, providing explicit formulas and geometric insights.

Contribution

It introduces a novel analysis of the expected crossing point for combined deterministic and random sequences, including explicit formulas and geometric interpretations.

Findings

Derived explicit formulas for the expected crossing point.

Provided geometric interpretations of the problem.

Connected results to known special cases.

Abstract

We consider the linear combinations of elements of two sequences: the first one a priory given nonnegative sequence and the second random sequence from the unit interval. We investigate the expected value of the smallest natural number such that the value of these linear combinations exceed a positive number. After very clear geometrical conclusions, we find the function which expresses the expected value. Here, we recognize a few known results like the special cases.