Chapter 3 Series

In this section, we shall study a particular kind of sequence called series. The terms of a series are formed by repeatedly adding together terms of some fixed sequence. For instance, if we start with the sequence $(1/k)_{k\in\mathbb{N}}$, then the corresponding series is given by

so the $n$th term is formed by summing the first $n$ terms of $(1/k)_{k\in\mathbb{N}}$. This particular example turns out to be important and interesting; it is called the harmonic series and we shall study it in detail.

Infinite series are fundamental across the whole breadth of the mathematical sciences. For instance, they will allow us to finally make complete sense of infinite decimal expansions such as $\sqrt{2}=1.4142\ldots$, something we discussed at the beginning of Chapter 2. Indeed, when we write $\sqrt{2}=1.4142\ldots$, we are intuitively encoding how to represent $\sqrt{2}$ in terms of powers of $10$: namely,

But what do the $\cdots$’s really mean here? A more precise understanding of the decimal expansion $\sqrt{2}=1.4142...$ is that it implicitly expresses $\sqrt{2}$ as the limit of the series formed from the sequence $(a_{n})_{n\in\mathbb{N}}$ where

Thus, series are fundamental to how we think about and represent real numbers. We shall explain these ideas in detail later in this chapter (see Definition 3.8).

Decimal expansions are one basic application of series, but you will see many other equally fundamental examples later in this course and in more advanced courses. These include the Taylor series expansion of a function, Fourier series and the construction of special functions such as the exponential function $\exp$.

Since series are so important, they come with their own special notation and terminology. We begin by introducing the basic definitions.

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  Establish the value of the limit of a given series using simple tools such as telescoping sums or the geometric series formula.

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  Apply convergence tests and limit laws for series to determine whether a given series converges or diverges.

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  Apply basic convergence tests (such as the boundedness test for series, $k$th term test and comparison test) to prove theoretical results about series.

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  Apply convergence tests to determine with a given series converges conditionally, converges absolutely or diverges.

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  Create examples to illustrate different possible behaviours of series.