2.9 Series

A series is what results from adding the terms of a sequence. The central question is whether an infinite sum of this kind has a finite value at all, and it is not settled by the terms merely shrinking: the terms of the harmonic series tend to zero while its sum grows without bound.

The subsections below build the answer in order. Sequences come first, since a series is defined as the limit of its sequence of partial sums. The convergence tests follow — each a different way of comparing an unfamiliar series with one whose behaviour is known. Power series then reverse the question: instead of asking whether a given series converges, they treat the sum as a function of \(x\) and ask where it is defined, which leads to the representation of familiar functions as infinite polynomials.

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