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
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.