The Rise and Development of the Theory of Series up to the Early 1820s

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ISBN: 978-0-387-73467-5 (Print) 978-0-387-73468-2 (Online)

Table of contents (33 chapters)

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  1. Front Matter

    Pages I-XV

  2. From the beginnings of the 17th century to about 1720: Convergence and formal manipulation

    1. Front Matter

      Pages 1-2

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      Book Chapter

      Pages 3-24

      Series before the rise of the calculus

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      Book Chapter

      Pages 25-44

      Geometrical quantities and series in Leibniz

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      Pages 45-51

      The Bernoulli series and Leibniz’s analogy

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      Pages 53-78

      Newton’s method of series

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      Pages 79-85

      Jacob Bernoulli’s treatise on series

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      Pages 87-92

      The Taylor series

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      Pages 93-113

      Quantities and their representations

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      Pages 115-120

      The formal-quantitative theory of series

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      Pages 121-130

      The first appearance of divergent series

  3. From the 1720s to the 1760s: The development of a more formal conception

    1. Front Matter

      Pages 131-132

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      Book Chapter

      Pages 133-140

      De Moivre’s recurrent series and Bernoulli’s method

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      Pages 141-146

      Acceleration of series and Stirling’s series

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      Pages 147-153

      Maclaurin’s contribution

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      Pages 155-169

      The young Euler between innovation and tradition

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      Pages 171-179

      Euler’s derivation of the Euler–Maclaurin summation formula

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      Pages 181-184

      On the sum of an asymptotic series

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      Pages 185-192

      Infinite products and continued fractions

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      Pages 193-199

      Series and number theory

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      Pages 201-214

      Analysis after the 1740s

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      Pages 215-229

      The formal concept of series

  4. The theory of series after 1760: Successes and problems of the triumphant formalism

    1. Front Matter

      Pages 231-231

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      Book Chapter

      Pages 233-237

      Lagrange inversion theorem

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