Sequences and Series— in which we discover very odd behaviour in even the smallest infinite set

Abstract

Much of the effort we have expended in developing the system ℝ of real numbers and in describing the infinite can now be put to good use. In this chapter we shall be investigating limiting processes, the very foundation of analysis. We begin by agreeing that an infinite sequence is a countably infinite set of real numbers occurring in some definite order, a₁,a₂,a₃, …, a _n , …. Each a _i ∈ℝ and there is one a _i for each i∈ℕ. A favoured abbreviation for a sequence is (a _n ), where a _n denotes the nth term of the sequence.

‘A quantity which is increased or decreased by an infinitely small quantity is neither increased nor decreased.’

Johann Bernouilli

‘How often have I said to you that when you have eliminated the impossible, whatever remains, however improbable, must be the truth.’

Sir Arthur Conan Doyle