series

**analysis** (differential/integral calculus, functional analysis, topology)

metric space, normed vector space

open ball, open subset, neighbourhood

convergence, limit of a sequence

compactness, sequential compactness

continuous metric space valued function on compact metric space is uniformly continuous

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

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 $b_k = \sum_{n=0}^k a_k$.

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.

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.