Series (summation)

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Created for functional analysis - work needed

Series

[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]

Definition

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]" 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 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.

This may be written:

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

Work

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