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.

The manuscript gives a coherent and detailed account of the theory of series in the eighteenth and early nineteenth centuries. It provides in one place an account of many results that are generally to be found - if at all - scattered throughout the historical and textbook literature. It presents the subject from the viewpoint of the mathematicians of the period, and is careful to distinguish earlier conceptions from ones that prevail today.