## The Rise and Development of the Theory of Series up to the Early 1820sThe theory of series in the 17th and 18th centuries poses several interesting problems to historians. Indeed, mathematicians of the time derived num- ous results that range from the binomial theorem to the Taylor formula, from the power series expansions of elementary functions to trigonometric series, from Stirling’s series to series solution of di?erential equations, from theEuler–MaclaurinsummationformulatotheLagrangeinversiontheorem, from Laplace’s theory of generating functions to the calculus of operations, etc. Most of these results were, however, derived using methods that would be found unacceptable today, thus, if we look back to the theory of series priortoCauchywithoutreconstructinginternalmotivationsandtheconc- tual background, it appears as a corpus of manipulative techniques lacking in rigor whose results seem to be the puzzling fruit of the mind of a - gician or diviner rather than the penetrating and complex work of great mathematicians. For this reason, in this monograph, not only do I describe the entire complex of 17th- and 18th-century procedures and results concerning series, but also I reconstruct the implicit and explicit principles upon which they are based, draw attention to the underlying philosophy, highlight competing approaches, and investigate the mathematical context where the series t- ory originated. My aim is to improve the understanding of the framework of 17th- and 18th-century mathematics and avoid trivializing the complexity of historical development by bringing it into line with modern concepts and views and by tacitly assuming that certain results belong, in some unpr- lematic sense, to a uni?ed theory that has come down to us today. |

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### Contents

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3 | |

2 Geometrical quantities and series in Leibniz | 25 |

22 Power series | 36 |

3 The Bernoulli series and Leibnizs analogy | 45 |

4 Newtons method of series | 53 |

41 The expansion of quantities into convergent series | 54 |

42 On Newtons manipulations of power series | 67 |

182 Functions relations and analytical expressions | 205 |

183 On the continuity of curves and functions | 211 |

19 The formal concept of series | 215 |

192 The impossibility of the quantitative approach | 219 |

193 Eulers deﬁnition of the sum | 222 |

The theory of series after 1760 Successes and problems of the triumphant formalism | 231 |

20 Lagrange inversion theorem | 233 |

21 Toward the calculus of operations | 239 |

5 Jacob Bernoullis treatise on series | 79 |

6 The Taylor series | 87 |

7 Quantities and their representations | 93 |

72 Continuous quantities numbers and ﬁctitious quantities | 100 |

8 The formalquantitative theory of series | 115 |

9 The ﬁrst appearance of divergent series | 121 |

From the 1720s to the 1760s The development of a more formal conception | 131 |

10 De Moivres recurrent series and Bernoullis method | 133 |

11 Acceleration of series and Stirlings series | 141 |

12 Maclaurins contribution | 147 |

13 The young Euler between innovation and tradition | 155 |

132 Analytical and synthetical methods in series theory | 160 |

133 The manipulation of the harmonic series and inﬁnite equations | 165 |

14 Eulers derivation of the EulerMaclaurin summation formula | 171 |

15 On the sum of an asymptotic series | 181 |

16 Infinite products and continued fractions | 185 |

17 Series and number theory | 193 |

18 Analysis after the 1740s | 201 |

22 Laplaces calculus of generating functions | 245 |

23 The problem of analytical representation of nonelementary quantities | 251 |

24 Inexplicable functions | 257 |

25 Integration and functions | 263 |

26 Series and differential equations | 267 |

27 Trigonometric series | 275 |

28 Further developments of the formal theory of series | 283 |

29 Attempts to introduce new transcendental functions | 297 |

30 DAlembert and Lagrange and the inequality technique | 303 |

The decline of the formal theory of series | 311 |

31 Fourier and Fourier series | 315 |

32 Gauss and the hypergeometric series | 323 |

33 Cauchys rejection of the 18thcentury theory of series | 347 |

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### Other editions - View all

The Rise and Development of the Theory of Series up to the Early 1820s Giovanni Ferraro No preview available - 2008 |

The Rise and Development of the Theory of Series up to the Early 1820s Giovanni Ferraro No preview available - 2010 |

### Common terms and phrases

18th-century 18th-century mathematicians algebraic analytical expression applied approximate arithmetica assumed asymptotic series Bernoulli binomial calculus Cauchy Cauchy’s century Chapter coefficients concept considered continued fractions convergent series curve d’Alembert Daniel Bernoulli defined definition denoted derived determined differential equation divergent series elementary functions equal Euler expansion finite fluxion formal formula Fourier function f(x Gauss geometric series geometrical quantities given infinite number infinite series infinitesimal instance integral interpolation interval intuition investigation Jacob Bernoulli Johann Bernoulli Lagrange Lagrange’s Laplace Leibniz GMS logarithm Maclaurin manipulation mathematics means method modern Moivre Newton notion numbers objects observed obtained operations Panza partial sums power series problem procedure proof Proposition quadratura recurrent series refer relation roots segment sense sequence series theory solution summation symbols Taylor series Taylor theorem term theorem theory of series tion trigonometric series valid variable quantity Vi`ete Wallis Wallis’s

### Popular passages

Page viii - Sufism, was revivified at the end of the 18th century and the beginning of the 19th century by Mulay al-'Arabi ad-Darqawi.