Editing Series (mathematics) (section)

===Convergence criteria===
The investigation of the validity of infinite series is considered to begin with [[Carl Friedrich Gauss|Gauss]] in the 19th century. Euler had already considered the hypergeometric series

<math display=block>1 + \frac{\alpha\beta}{1\cdot\gamma}x + \frac{\alpha(\alpha+1)\beta(\beta+1)}{1 \cdot 2 \cdot \gamma(\gamma+1)}x^2 + \cdots</math>

on which Gauss published a memoir in 1812. It established simpler criteria of convergence, and the questions of remainders and the range of convergence.

[[Cauchy]] (1821) insisted on strict tests of convergence; he showed that if two series are convergent their product is not necessarily so, and with him begins the discovery of effective criteria. The terms ''convergence'' and ''divergence'' had been introduced long before by [[James Gregory (astronomer and mathematician)|Gregory]] (1668). [[Leonhard Euler]] and [[Carl Friedrich Gauss|Gauss]] had given various criteria, and [[Colin Maclaurin]] had anticipated some of Cauchy's discoveries. Cauchy advanced the theory of [[power series]] by his expansion of a complex [[function (mathematics)|function]] in such a form.

[[Niels Henrik Abel|Abel]] (1826) in his memoir on the [[binomial series]]

<math display=block>1 + \frac{m}{1!}x + \frac{m(m-1)}{2!}x^2 + \cdots</math>

corrected certain of Cauchy's conclusions, and gave a completely scientific summation of the series for complex values of <math>m</math> and <math>x</math>. He showed the necessity of considering the subject of continuity in questions of convergence.

Cauchy's methods led to special rather than general criteria, and
the same may be said of [[Joseph Ludwig Raabe|Raabe]] (1832), who made the first elaborate investigation of the subject, of [[Augustus De Morgan|De Morgan]] (from 1842), whose
logarithmic test [[Paul du Bois-Reymond|DuBois-Reymond]] (1873) and [[Alfred Pringsheim|Pringsheim]] (1889) have
shown to fail within a certain region; of [[Joseph Louis François Bertrand|Bertrand]] (1842), [[Pierre Ossian Bonnet|Bonnet]]
(1843), [[Carl Johan Malmsten|Malmsten]] (1846, 1847, the latter without integration); [[George Gabriel Stokes|Stokes]] (1847), [[Paucker]] (1852), [[Chebyshev]] (1852), and [[Arndt]]
(1853).

General criteria began with [[Ernst Kummer|Kummer]] (1835), and have been studied by [[Gotthold Eisenstein|Eisenstein]] (1847), [[Weierstrass]] in his various
contributions to the theory of functions, [[Ulisse Dini|Dini]] (1867),
DuBois-Reymond (1873), and many others. Pringsheim's memoirs (1889) present the most complete general theory.