Sequences and Series

1. About these pages

These web pages present a web-based first course in real analysis (part of what is covered at university level in the USA as calculus though this terminology does not apply in the UK and the notes here follow the UK style). In particular these pages cover the notions of convergence of sequences and series and the nature of the real numbers. Content is provided by a mixture of HTML and MathML. All content here may be copied as often as you like under the terms of the Gnu Free Documentation Licence.

These web pages were originally written to support the modules MSM1Bb and MSM2P01b given in the School of Mathematics, University of Birmingham, from 2005-6 onwards, for students taking single honours or combined/joint honours programmes in Mathematics. They have since been updated for subsequent years, and now contain rather more material than is required for this module, or other similar modules.

The pages will form a useful or main source of information for students taking this module or a similar module and may be of interest to many other people too for mathematical reasons. They also provide a working example of the Gloss system for authoring HTML and MathML. The pages were updated recently (2015) to take advantage of MathJax and a new version of Gloss.

The author is Dr Richard Kaye, Email:

All of the web pages with URI starting (which may for convenience also be found on other locations in other versions) are written and copyright by Richard Kaye.

They may be copied under the terms of the Gnu Free Documentation Licence. This means that you may download, copy and modify them at will, and make your own private version. If you distribute this private version or indeed any work at all derived from these pages you must distribute it under an identical or compatible licence. In particular you may not charge for your work save a reasonable charge for transmission costs or electronic media.

Almost all these pages contain mathematics in MathML format, and are rendered in your browser using javascript. More details are here.

Please email me at R.W.Kaye[at] if you find these notes useful, or have any other comment on them.

2. Course content

The mathematical content in these pages is a typical first course in real analysis of the type usually given in first or second year at a British university. The main idea being developed here is the idea of a limit of a sequence of numbers or of an infinite series of numbers. We can think of a sequence or series as an infinite process that may (or may not) converge to a value, and the first step is to define the correct notion of convergence. The ideas are particularly rich and there are many things one can say about convergence in general, and some striking examples. A related subsidiary idea in these notes is the nature of the real numbers themselves, because these numbers can be characterised exactly by the way they behave under the concept of limit. It is often a surprise to beginners to the subject that the main concept in mathematics - number - is secondary to a somewhat trick topic in first year analysis.

As a result, the ideas of convergence studied here will be enough to make precise the nature of the real numbers, integration, differentiation, and many functions such as exponentiation and trigonometric functions, all of which need the ideas of convergence and limits to be defined precisely. (It is often a surprise to students that the definitions of these apparently familiar ideas require such sophisticated work.)

Because it is easy to make logical errors and fall into paradoxes it is important to be very precise and rigorous in doing this work. In fact, a major part of these notes is an introduction to mathematical rigour and abstract mathematics, and this in any case forms an important part of any first-year pure mathematics course. So these notes also include a certain amount of material about logic, proofs and quantifiers, much of which is unfamiliar to students beginning this topic.

3. Licence and copyright

Preamble: In the following Licence Notice, this document refers to all web pages and other electronic documents available via http at any of the addresses, or a subdirectory of one of these, or a copy of one of these. All such content, lecture notes or other materials taken from these web pages are being made freely available under the Gnu Free Documentation Licence (see That means you may copy it freely for any purpose you like provided that if any part is incorporated in a work that you write or publish, then that work of yours must be available freely under a similar licence. (If you do make use of these notes, in whatever form, I would be glad to hear from you and would be grateful if my work is attributed correctly in yours.) The legal details are all at There is no warranty of any kind on these web pages.

Copyright (c), 2006,2015, Richard Kaye, Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 at or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts.

4. Table of Contents

When you have read the information above, including the licence, and are ready to do so, please click the following link to take you to the Table of Contents.