The Summation of Series

This is a broad but not very deep look at a number of topics related to the calculus of finite differences. The series of the title can be finite or infinite sums, with the emphasis on the finite ones. The treatment can aim at either a closed-form expression or a numerical approximation.

The book explains a wide variety of methods, ranging from high-school level (interpolation from tables, sometimes using up to 3^rd differences), through advanced undergraduate (Euler–Maclaurin sum formula, Poisson summation formula, asymptotic series). The numerical work generally omits error analysis, instead guessing when we have done enough by looking at the sizes of the difference terms when interpolating or at the size of the first omitted term when using series.

The present volume is a 2015 unaltered reprint of a 1962 work and although not obsolete, it is archaic. We don’t look at these subjects the same way today that we did back in 1962, largely due to the influence of computers. A good modern introduction to finite differences is Chapter 2 of Graham & Knuth & Patashnik’s Concrete Mathematics. The present work has almost nothing about sums with binomial coefficients, because there was no systematic way to handle them then, but that too has changed with the invention of the Gosper algorithm and the Wilf-Zeilberger algorithm. There is a good introduction to these subjects also in Chapter 5 of Concrete Mathematics, as well as an entire book on the subject, A=B by Petkovsek & Wilf & Zeilberger. The numerical portions of the present volume are not very advanced, but they would be covered in any present-day work on numerical analysis.

What’s left? The present volume is still full of useful worked-out examples. I would not recommend it as a text, because it gives a distorted view of the field today. It has some value as a reference, with a fairly lengthy table of finite sums, although there are much more comprehensive books available today, as well as computer-algebra systems that can work out many sums. Gradshteyn & Ryzhik’s Table of Integrals, Series, and Products is still being updated and, although primarily an integrals book, does have extensive sections on finite and infinite sums. Another excellent table, although out-of-print today, is L. B. W. Jolley’s Summation of Series (Dover, 2^nd revised edition, 1961).

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.