## 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. |

### Contents

1 | |

Geometrical quantities and series in Leibniz | 25 |

The Bernoulli series and Leibnizs analogy | 45 |

Newtons method of series | 53 |

Jacob Bernoullis treatise on series | 79 |

The Taylor series | 87 |

Quantities and their representations | 93 |

The formalquantitative theory of series | 115 |

Analysis after the 1740s | 201 |

The formal concept of series | 215 |

Successes | 231 |

Toward the calculus of operations | 239 |

Laplaces calculus of generating functions | 245 |

The problem of analytical representation | 251 |

Inexplicable functions | 257 |

Integration and functions | 263 |

The first appearance of divergent series | 121 |

The development | 131 |

Acceleration of series and Stirlings series | 141 |

Maclaurins contribution | 147 |

The young Euler between innovation and tradition | 155 |

Eulers derivation of the EulerMaclaurin | 171 |

On the sum of an asymptotic series | 181 |

Series and number theory | 193 |

Trigonometric series | 275 |

Further developments of the formal theory of series | 283 |

Attempts to introduce new transcendental functions | 297 |

The decline of the formal theory of series | 311 |

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

363 | |

383 | |

### 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 - 2011 |

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

### Common terms and phrases

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### 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.