definition of series
Topics
About: ring
The table of contents of this article
Starting Context
- The reader knows a definition of ring.
- The reader knows a definition of sequence.
Target Context
- The reader will have a definition of series.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( J\): \(\subseteq \mathbb{N}\)
\( R\): \(\in \{\text{ the rings }\}\)
\( s\): \(\in \{\text{ the sequences }\}\), such that \(Dom (s) = J\) and \(Ran (s) \subseteq R\)
\(*\widetilde{s}\): \(= \sum_{j \in \{1, 2, ...\}} s_j\)
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Conditions:
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\(\widetilde{s}\) is often denoted as \(\sum_{j \in J} s (j)\), where the sum is understood to be taken in the increasing order
2: Note
In general, \(\widetilde{s}\) is just the entity denoted as \(\sum_{j \in \{1, 2, ...\}} s_j\), not necessarily any point on \(R\); when \(J\) is finite, \(\widetilde{s}\) is identified with the point on \(R\); when \(J\) is infinite and \(R\) is a metric space, \(\widetilde{s}\) may converge to \(r \in R\), and then, \(\widetilde{s}\) is identified with \(r\).
