structural:summarization
Convergence criteria

The investigation of the validity of infinite series is considered to begin with Gauss in the 19th century. Euler had already considered the hypergeometric series

1 + α β 1 ⋅ γ x + α (α + 1) β (β + 1) 1 ⋅ 2 ⋅ γ (γ + 1) x 2 + ⋯ {\displaystyle 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 }

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 Gregory (1668). Leonhard Euler and 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 in such a form.

Abel (1826) in his memoir on the binomial series

1 + m 1 ! x + m (m − 1) 2 ! x 2 + ⋯ {\displaystyle 1+{\frac {m}{1!}}x+{\frac {m(m-1)}{2!}}x^{2}+\cdots }

corrected certain of Cauchy's conclusions, and gave a completely scientific summation of the series for complex values of m {\displaystyle m} and x {\displaystyle x}. He showed the necessity of considering the subject of continuity in questions of convergence.

Cauchy's methods led to special rather than general criteria, andthe same may be said of Raabe (1832), who made the first elaborate investigation of the subject, of De Morgan (from 1842), whoselogarithmic test DuBois-Reymond (1873) and Pringsheim (1889) haveshown to fail within a certain region; of Bertrand (1842), Bonnet (1843), Malmsten (1846, 1847, the latter without integration); Stokes (1847), Paucker (1852), Chebyshev (1852), and Arndt (1853).

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