This is a preview of subscription content, log in via an institution to check access.
Additional Bibliography
Andersen, A. F., Bemerkung zum Beweis des Herrn Knopp für die Äquivalenz der Cesàro- und Hölder-Summabilität. Math. Zeit., 28, 356–359 (1928).
Borel, E., Applications de la théorie des séries divergentes sommables. Comptes Rendus, 122, 805–807 (1896).
Borel, E., Fondements de la théorie des séries divergentes sommables. Journal de Math., (5) 2, 103–133 (1896).
Borel, E., Le prolongement analytique et les séries sommables. Math. Annalen, 55, 74–80 (1901).
Borel, E., Mémoire sur les séries divergentes. Ann. Sci. de l'Ecole Norm. Sup., (3) 16, 9–136 (1899).
Bromwich, T. J., On the limits of certain infinite series and integrals. Math. Annalen, 65, 313–349 (1908).
Buhl, A., Application du procédé de sommation de M. E. Borel aux séries trigonométriques généralisees. Comptes Rendus, 143, 445–446 (1906).
Buhl, A., Sur de nouvelles formules de sommabilité. Bull. des Sciences, 42, 340–346 (1907).
Buhl, A., Sur la sommabilite des séries d'une variable réélle ou complexe. Journal de Math., (6) 4, 367–393 (1908).
Cajori, F., A History of Mathematics. New York: Macmillan 1919.
Cantor, M., Vorlesungen über Geschichte der Mathematik. 4 Vols. Leipzig: Teubner, 1900–1908.
Carmichael, R. D., General aspects of the theory of summable series. Bull. of the Amer. Math. Soc., 25, 97–131 (1919).
Cauchy, A., Sur un emploi légitime des séries divergentes. Comptes Rendus, 17, 370–376 (1843).
Cesàro, E., Contributions à la théorie des limites. Bull. des Sciences, (2) 13, 51–54 (1899).
Cesàro, E., Nuova contribuzione ai principii fondamentali dell' aritmetics assintotica. Rendiconti Napoli, (2) 7, 208–209 (1893).
Chapman, S., A note on the theory of summable integrals. Bull. of the Amer. Math. Soc., 18, 111–117 (1911).
Chapman, S., On the summability of series of Legendre's functions. Math. Annalen, 72, 211–227 (1912).
Chessin, A. S., On divergent series. Bull. of the Amer. Math. Soc., (2) 2, 72–75 (1895).
Dienes, P., La série de Taylor sur le cercle de convergence. Comptes Rendus, 140, 489–491 (1905).
Ford, W. B., A set of criteria for the summability of divergent series. Bull. of the Amer. Math. Soc., (2) 15, 439–444 (1909).
Ford, W. B., On the relation between the sum formulas of Hölder and Cesàro. Amer. Journal of Math., 32, 315–326 (1910).
Ford, W. B., A conspectus of the modern theory of divergent series. Bull. of the Amer. Math. Soc., 25, 1–15 (1918).
Ford, W. B., Studies on Divergent Series and Summability. New York: Macmillan 1916.
Hardy, G. H., An extension of a theorem on oscillating series. Proc. of the London Math. Soc., 12, 156–173 (1913).
Hardy, G. H., Further researches in the theory of divergent series and integrals. Trans. of the Cambr. Phil. Soc., 21, 1–48 (1908).
Hardy, G. H., Generalization of a theorem in the theory of divergent series. Proc. of the London Math. Soc., (2) 6, 225–264 (1908).
Hardy, G. H., A note on divergent Fourier series. Mess. of Math., (2) 33, 137–144 (1904).
Hardy, G. H., & S. Chapman, A general view of the theory of summable series. Quar. Journal of Math., 42, 181–215 (1911).
Hellinger, E., & O. Toeplitz, Grundlagen für eine Theorie der unendlichen Matrizen. Göttingen Nach., 351–355 (1906).
Hurwitz, W. A., Report on topics in the theory of divergent series. Bull. of the Amer. Math. Soc., 28, 17–36 (1922).
James, G., On the theory of summability. Annals of Math., (2) 21, 120–127 (1919).
Klein, F., Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert. Berlin, 1926.
Knopp, K., Über das Eulersche Summierungsverfahren. Math. Zeit., 15, 226–253 (1922).
Le Roy, E., Sur les séries divergentes et les fonctions définies par un développement de Taylor. Annales de Toulouse, (2) 2, 317–430 (1900).
Lindelöf, E., Quelques applications d'une formule sommatoire générale. Acta Soc. Fennicae, 31, 1–46 (1902).
Perron, O., Beitrag zur Theorie der divergenten Reihen. Math. Zeit., 6, 286–310 (1920).
Poisson, S. D., Mémoire sur le calcul numérique des intégrales définies. Mémoires Acad. Sci. Inst. France 6, 571–602 (1823).
Pringsheim, A., Divergente Reihen. Ency. d. math. Wissenschaften, IA 3, Secs. 38–40: 1198.
Ricotti, E., Sulle serie divergenti sommabili. Giornale di Mat. di Battaglini, 48, 79–111 (1910).
Riemann, B., Gesammelte Mathematische Werke. New York: Dover reprint, 1953.
Riesz, M., Sur les séries de Dirichlet et les séries entières. Comptes Rendus, 149, 909–912 (1909).
Schur, I., Über die Äquivalenz der Cesàroschen und Hölderschen Mittelwerte. Math. Annalen, 74, 447–458 (1913).
Silverman, L. L., On the definition of the sum of a divergent series. Univ. of Missouri Studies, 1–96, 1913.
Smail, L. L., History and Synopsis of the Theory of summable Infinite Processes. Eugene, Oregon: Univ. Press 1925.
Takenaka, S., A general view of the theory of summability. Tohoku Math. Journal, 21, 193–221 (1922).
Whittaker, E. T., A Course of Modern Analysis. Cambridge: Univ. Press 1902.
Author information
Authors and Affiliations
Additional information
Communicated by M. Kline
Rights and permissions
About this article
Cite this article
Tucciarone, J. The development of the theory of summable divergent series from 1880 to 1925. Arch. Hist. Exact Sci. 10, 1–40 (1973). https://doi.org/10.1007/BF00343405
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00343405