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.