This is a rambling look at the concept and use of infinite series through the ages. The scholarship is good but the writing is poor.

The modern rigorous definition of the sum of an infinite series was not nailed down until Cauchy and Weierstrass in the mid-1800s. Despite this, series had been used for hundreds of years (thousands, if we count the ancient Greeks), in particular by Newton and by Euler. They produced many useful and mostly-correct results, even though by modern standards their proofs were inadequate. The book attempts to show how infinite series were viewed and conceptualized by these mathematicians. It generally gives examples of their results, along with some of their reasoning. As we get nearer the present the book starts stating and proving theorems, but it’s not a textbook on infinite series.

The coverage starts with the method of exhaustion and other infinite processes studied by the ancient Greeks. The bulk of the book deals with the results of Newton and Euler, and with Cauchy’s later systemization of limits. There’s also a good bit on the later rigorizations due to Weierstrass, Bolzano, Dirichlet, Heine, and Dedekind. The book pays particular attention to trigonometric series and Fourier series and contrasts them with power series. The final chapter looks beyond convergence to summability.

The author is German, writing in English, and the language is unidiomatic and usually rough going. Some samples: “This famous accomplishment, gained by an essential of his Calculus, opened Leibniz’ career as a mathematician.” (p. 23; there’s probably a word missing after “essential). “Cauchy talked approach all the way.” (p. 69) “Newton, Leibniz, Euler, Cauchy having played kick and hope with infinite series, luckily compare to a soccer team in that their offside goals subsequently were confirmed by uniform convergence.” (p. 77) “The career of inverse theorems started when the Austrian Alfred Tauber wrote one and only one paper, enough to render his name immortal in summability after Hardy had coined ‘Tauberism’.” (p. 108)

A few German terms sneaked in untranslated or half-translated, for example arcustangent and polynom. There are a few typos, some (very unfortunately for a history book) in the dates: the book has Cauchy publishing a result in 1921 and the present-day mathematician Korevaar publishing a book in 1668. There are a few works that are cited in the body but not listed in the references, such as several by a mysterious author “Sonar” who is probably the German mathematician and historian Thomas Sonar. One harmless peculiarity of the references is that they often cite both works originally written in English and their German translations, and the footnotes give the page references for both.

Bottom line: The book has many interesting things, but I don’t recommend because it is so hard to read. You are better off with one of the more general histories of calculus, such as Edward’s *The Historical Development of the Calculus* or Boyer’s older *The History of the Calculus and Its Conceptual Development*.

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.