  Sequences and Series An introduction to mathematical analysis Convergence, limits, etc. and the foundations of differential and integral calculus. back to home page

Here is my article on Fourier series and transforms, including the Fast Fourier Transform (FFT) algorithm).

Here is the article on summing powers of an integer, Bernoulli numbers and the Euler-Maclaurin summation formula.

This section includes the full text of two textbooks by Dr. John Reade of the Maths Department, University of Manchester.

'An Introduction to Mathematical Analysis’ by John Reade was first published in 1986 by Oxford University Press as a undergraduate textbook for a first course in formal analysis. It is very readable yet thorough, and remains a very helpful broad introduction to the subject. It is published here in full by permission of the author and OUP, from the pages of the OUP book.

‘Uniform Convergence ’ by John Reade has not previously been published. The book is a university second level text on the convergence of infinite series of functions, fn(x). Uniform convergence is more powerful than mere pointwise convergence – for example, series of functions which are uniformly convergent can be differentiated term by term, and their limiting behaviour is stable. Historically, uniform convergence became recognised as explaining why the newly discovered Fourier series could describe discontinuous functions.

Download 'An Introduction to Mathematical Analysis’ as individual chapters in pdf format:

•     Preface and contents
•     Chapter 1 : Definition and illustration of uniform convergence.
•     Chapter 2 : Analytic properties of uniform limits. 'Almost' uniform convergence.
•     Chapter 3 : The Cauchy criterion. Dirichlet and Abel tests for conditional convergence.
•     Chapter 4 : Uniform convergence of series of functions. Weiestrass’s M-Test.
•     Chapter 5 : Power series. Multiplication of series. Rigorous proof of the binomial theorem.
•     Chapter 6 : Fourier series. Rigorous treatment of the convergence properties of Fourier series. Cesaro summability of Fourier series of continuous functions.
•     Appendix : Fourier series of continuous functions.

The diagrams for Uniform Convergence are still being prepared.

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