Table of Contents

From the beginnings of the 17th century to about 1720: Convergence and formal manipulation.- Series before the rise of the calculus.- Geometrical quantities and series in Leibniz.- The Bernoulli series and Leibniz’s analogy.- Newton’s method of series.- Jacob Bernoulli’s treatise on series.- The Taylor series.- Quantities and their representations.- The formal-quantitative theory of series.- The first appearance of divergent series.- From the 1720s to the 1760s: The development of a more formal conception.- De Moivre’s recurrent series and Bernoulli’s method.- Acceleration of series and Stirling’s series.- Maclaurin’s contribution.- The young Euler between innovation and tradition.- Euler’s derivation of the Euler–Maclaurin summation formula.- On the sum of an asymptotic series.- Infinite products and continued fractions.- Series and number theory.- Analysis after the 1740s.- The formal concept of series.- The theory of series after 1760: Successes and problems of the triumphant formalism.- Lagrange inversion theorem.- Toward the calculus of operations.- Laplace’s calculus of generating functions.- The problem of analytical representation of nonelementary quantities.- Inexplicable functions.- Integration and functions.- Series and differential equations.- Trigonometric series.- Further developments of the formal theory of series.- Attempts to introduce new transcendental functions.- D’Alembert and Lagrange and the inequality technique.- The decline of the formal theory of series.- Fourier and Fourier series.- Gauss and the hypergeometric series.- Cauchy’s rejection of the 18th-century theory of series.

Reviews

From the reviews: "Giovanni Ferraro’s book must be regarded as an important contribution to the history of mathematical analysis … . In summary, this book is thoroughly researched; it is written with a high degree of accuracy, and the broad range of fascinating material is, in general, very well organized. Therefore … every college and university library should have a copy. In fact, extracts from the book could be used to inject historical perspectives into courses on analysis, and it will certainly appeal to historians in general." (Peter Ruane, MathDL, April, 2008) "The work is thoroughly referenced and should prove valuable to mathematical history researchers … . Summing Up: Recommended. Researchers/faculty and professionals." (C. Bauer, CHOICE, Vol. 46 (01), September, 2008) "The author begins his detailed account with the arrival of the calculus with Newton and Leibniz, and their own treatments of series. … The range of series and functions treated is impressive … . The story stops with a brief notice of the start of the critical phase, especially concerning the convergence (or not) of infinite series, which was put forward by Cauchy and his followers … . The book is a valuable contribution, possibly the single most substantial source on the topic." (I. Grattan-Guinness, Mathematical Reviews, Issue 2008 k)