nLab
series

Motivation

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

Definition

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

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

• geometric series

• Taylor series

Related concepts

• asymptotic series

• infinite product

• divergent series

category: analysis