nLab
series

Infinite series

Motivation

A series is a formal precursor to a various notions of a sum of an infinite sequence.

Definition

An (ordinary) series n=0 a n\sum_{n=0}^\infty a_n whose members are the elements a na_n in a given additive group is an ordered pair of a sequence (a n) n=0 (a_n)_{n=0}^\infty and a sequence (b k) k(b_k)_k of its partial sums b k= n=0 ka kb_k = \sum_{n=0}^k a_k.

Sum of a series

The most straightforward notion of the sum of a series is the limit of its sequence of partial sums, if this sequence converges, relative to some topology on the space where the members of the sequence belong to. A series that does not converge in this sense is called divergent; sometimes these can also be “summed” by fancier techniques.

Examples

category: analysis

Last revised on July 5, 2017 at 05:02:28. See the history of this page for a list of all contributions to it.