RePoSS #11: The Mathematics of Niels Henrik Abel: Continuation ...

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274 Chapter 13. ABEL and OLIVIER on convergence tests On the other hand, ABEL next turned to a function intimately related to the one studied above, 19 ψ (n) = φm (n) 1−α . 1 − α This time, ABEL’S calculations produced the inequality ψ (n + 1) − ψ (n) > ψ ′ (n + 1) corresponding to the fact that ψ ′ was a decreasing function. Consequently, through a number of calculations, ABEL was led to a series ∞ ∑ n=1 1 n log m (n) α+1 ∏ m−1 k=1 logk n which was convergent if α > 0 and another one (corresponding to α = −1) which was divergent. ∞ ∑ n=1 1 n ∏ m−1 k=1 logk n A logarithmic test of convergence. These methods of constructing convergent and divergent series led ABEL to a new test of convergence. The underlying idea of ABEL’S argument starts from the two series, one convergent and the other divergent, and compares a given series with these two typical ones. He found by simple arguments based on the results above, that if � log lim log m+1 n 1 unn ∏ m−1 k=1 logk n � > 1, (13.5) the series ∑ un was convergent. ABEL’S criterion also indicated, that if the limit in (13.5) was < 1, the series ∑ un would be divergent. In its polished form, ABEL’S criterion thus became the following: Theorem 15 For a series of positive terms ∑ un, the limit � 1 log un k = lim n→∞ d dn logm � n log m+1 n is considered. If k > 1, the series will be convergent; if k < 1, it will be divergent; and if k = 1, nothing can be said of the convergence or divergence of the series by this test. ✷ This result was later rediscovered by J. L. F. BERTRAND (1822–1900). 20 19 ABEL actually also denoted this function by φ, but to avoid confusion, I have chosen to label it ψ. 20 (Bertrand, 1842).
13.4. More characterizations and tests of convergence 275 ABEL’S continued interest in the theory of series and — in particular — in obtain- ing new tests of convergence for series are indications of a continued interest in this topic. Due to CAUCHY’S re-founding of analysis, tests of convergence were becoming increasingly important, and a number of new tests were discovered in the nineteenth century. 21 21 See e.g. (I. Grattan-Guinness, 1970b, 131–151).
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RePoSS: Research Publications on Sc
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The Mathematics of NIELS HENRIK ABE
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The Mathematics of NIELS HENRIK ABE
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Contents Contents i List of Tables
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8.3 Refocusing on the equation . .
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V ABEL’s mathematics and the rise
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List of Figures 2.1 NIELS HENRIK AB
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List of Boxes 1 The algebraic reduc
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Summary The present PhD dissertatio
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Preface to the 2004 edition For thi
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in connection with the Abel centenn
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items published in the same year ar
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the Mittag-Leffler archives in Djur
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Chapter 1 Introduction In the after
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1.2. The mathematical topics involv
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1.3. Themes from early nineteenth-c
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1.4. Reflections on methodology 9 l
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1.4. Reflections on methodology 11
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18 Chapter 2. Biography of NIELS HE
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Chapter 3 Historical background The
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3.2. ABEL’s position in mathemati
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3.3. The state of mathematics 43 tr
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3.4. ABEL’s legacy 45 As can be s
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Chapter 4 The position and role of
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4.1. Outline of ABEL’s results an
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4.2. Mathematical change as a histo
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4.2. Mathematical change as a histo
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58 Chapter 5. Towards unsolvable eq
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98 Chapter 6. Algebraic insolubilit
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100 Chapter 6. Algebraic insolubili
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Chapter 7 Particular classes of equ
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7.1. Solubility of Abelian equation
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7.2. Elliptic functions 153 Figure
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7.2. Elliptic functions 155 7.2.1 T
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7.3. The concept of irreducibility
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7.3. The concept of irreducibility
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7.4. Enlarging the class of solvabl
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164 Chapter 8. A grand theory in sp
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Part III Interlude: ABEL and the
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192 Chapter 9. The nineteenth-centu
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194 Chapter 10. Toward rigorization
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208 Chapter 11. CAUCHY’s new foun
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Chapter 12 ABEL’s reading of CAUC
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Page 261 and 262: 12.3. Convergence 231 and {εm} was
Page 263 and 264: 12.4. Continuity 233 it followed th
Page 265 and 266: 12.4. Continuity 235 Actually, if t
Page 267 and 268: 12.4. Continuity 237 DIRICHLET intr
Page 269 and 270: 12.5. ABEL’s “exception” 239
Page 271 and 272: 12.6. A curious reaction: Lehrsatz
Page 277 and 278: 12.7. From power series to absolute
Page 279 and 280: 12.7. From power series to absolute
Page 281 and 282: 12.8. Product theorems of infinite
Page 283 and 284: 12.8. Product theorems of infinite
Page 285 and 286: 12.9. ABEL’s proof of the binomia
Page 291 and 292: 12.10. Aspects of ABEL’s binomial
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Page 307 and 308: Chapter 14 Reception of ABEL’s co
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Page 311 and 312: 14.2. Conclusion 281 of basic notio
Page 313: Part IV Elliptic functions and the
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Page 353 and 354: 17.1. Infinite representations 323
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17.2. Elliptic functions as ratios
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17.2. Elliptic functions as ratios
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17.3. Characterization of ABEL’s
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Chapter 18 Tools in ABEL’s resear
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18.1. Transformation theory 333 The
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18.1. Transformation theory 335 Obv
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18.1. Transformation theory 337 Sum
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18.2. Integration in logarithmic te
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18.3. Conclusion 345 Summary. ABEL
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Chapter 19 The Paris memoir N. H. A
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19.1. ABEL’s approach to the Pari
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19.2. The contents of ABEL’s Pari
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19.3. Additional, tentative remarks
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19.3. Additional, tentative remarks
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19.4. The fate of the Paris memoir
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19.6. Conclusion 377 hyperelliptic
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Chapter 20 General approaches to el
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20.2. Other ways of introducing ell
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20.3. Conclusion 383 All these four
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Chapter 21 ABEL’s mathematics and
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21.1. From formulae to concepts 389
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21.1. From formulae to concepts 391
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21.2. Concepts and classes enter ma
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21.3. The role of counter examples
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21.4. Conclusion 401 It is beyond t
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404 Appendix A. ABEL’s correspond
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Appendix B ABEL’s manuscripts Man
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1. Précis d’une théorie des fon
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Bibliography Abel (MS:351:A). “M
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Bibliography 413 in: Journal für d
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Bibliography 415 — (1990). “Geo
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Bibliography 417 — (1830). “Mé
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Bibliography 419 — (1768). “Ins
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Bibliography 421 Grattan-Guinness,
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Bibliography 423 Jahnke, H. N. (198
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Bibliography 425 Legendre, A. M. (1
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Bibliography 427 Rosen, M. (1981).
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Bibliography 429 Toti Rigatelli, L.
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432 Index of names Frobenius, Georg
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Index École Normale, 40, 187 Écol
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Index 437 Lagrange interpolation, 3
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