Infinite series

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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
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.

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