The Basic Concepts of Infinite Series

Whether an infinite series converges is one of the few questions in mathematics where the answer is definite but hard to decide. This chapter establishes the most basic language and criteria.

The definition of convergence is that the sequence of partial sums converges to a finite value — and this definition deserves attention: judging convergence ultimately comes down to “does a certain sequence converge”, which is exactly what separates it from “adding infinitely many numbers together”. The concepts of partial sums, limits of sequences, convergence and divergence have to be precise, or none of the later tests mean anything.

This chapter also gives the necessary condition for convergence (the general term must tend to zero) along with the basic properties. It only rules out some divergent series — it is not enough to decide convergence — and that is precisely what the tests in the next chapter are for.