Series (summation)

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Series

$$\sum^\infty_{n\eq 1}x_n:\eq\lim_{n\rightarrow\infty}$$ $\Big(\sum^n_{i\eq 1}x_i\Big)$ $\big((x_n)_{n\in\mathbb{N} },(S_n)_{n\in\mathbb{N} }\big)$

For a sequence $(x_n)_{n\in\mathbb{N} }\subseteq X$ in a metric space $(X,d)$ that is also a group $(X,+)$

Definition

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

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

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

• The series, $\big((x_n)_{n\in\mathbb{N} },(S_n)_{n\in\mathbb{N} }\big)$, converges.

This may be written:

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

Work

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