Formation of averages: a classical process

Abstract: 

The formation of the average of a finite set of numbers crops up in various aspects of daily life; eg. the average daily temperature (say, at noon) for a given month, the average age of the members in a mathematics department (say, in August 2016), etc. Perhaps mathematically more interesting is to take an infinite set of numbers a1,a2,a3,⋯a1,a2,a3,⋯ and examine the behaviour of its sequence of averages, i.e., the sequence

a1,a1+a22,a1+a2+a33,⋯.a1,a1+a22,a1+a2+a33,⋯.

The purpose of this survey talk is to expose some of the properties of the “averaging process” which assigns to a given sequence (an)∞n=1(an)n=1∞ its sequence of averages 1n(∑nk=1ak)1n(∑k=1nak), where the initial sequence (an)∞n=1(an)n=1∞ is constrained to remain within a certain fixed class of sequences specified in advance. The talk is aimed at a mixed mathematical audience and should be largely accessible to honours students and beyond. Technicalities will be kept to a minimum.