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

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Springer Science & Business Media, Dec 20, 2007 - Mathematics - 392 pages
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The 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

From the beginnings of the 17th century to about 1720 Convergence and formal manipulation
1
1 Series before the rise of the calculus
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
References
363
Author Index
383
Subject Index
387
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