Series (summation)

From Maths
Jump to: navigation, search
Stub grade: A
This page is a stub
This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:
Created for functional analysis - work needed
[math]\sum^\infty_{n\eq 1}x_n:\eq\lim_{n\rightarrow\infty} [/math][ilmath]\Big(\sum^n_{i\eq 1}x_i\Big)[/ilmath]

[ilmath]\big((x_n)_{n\in\mathbb{N} },(S_n)_{n\in\mathbb{N} }\big)[/ilmath]
For a sequence [ilmath](x_n)_{n\in\mathbb{N} }\subseteq X[/ilmath]
in a metric space [ilmath](X,d)[/ilmath]
that is also a group [ilmath](X,+)[/ilmath]


Let [ilmath](X,d)[/ilmath] be a metric space such that [ilmath](X,+)[/ilmath] is a group. Let [ilmath](x_n)_{n\in\mathbb{N} }\subseteq X[/ilmath] be any sequence in [ilmath]X[/ilmath]. Then:

  • [ilmath]\big((x_n)_{n\in\mathbb{N} },(S_n)_{n\in\mathbb{N} }\big)[/ilmath] is a "series with general term [ilmath]x_n[/ilmath]"[1] where:
    • for all [ilmath]n\in\mathbb{N} [/ilmath] we may define [ilmath]S_n:=\sum^n_{i=1}x_i[/ilmath], this is called the [ilmath]n[/ilmath]th partial sum[1] of the series

If [ilmath](s_n)_{n\in\mathbb{N} } [/ilmath] converges (in the usual sense for sequences) then we may say:

  • The series, [ilmath]\big((x_n)_{n\in\mathbb{N} },(S_n)_{n\in\mathbb{N} }\big)[/ilmath], converges[1].

This may be written:

  • [ilmath]\sum^\infty_{n\eq 1}x_n<\infty[/ilmath][1] but this is an abuse of notation.


Grade: A
This page requires some work to be carried out
Some aspect of this page is incomplete and work is required to finish it
The message provided is:
Working on it right now


  1. 1.0 1.1 1.2 1.3 Analysis - Part 1: Elements - Krzysztof Maurin