Loterre: Mathematics: convergent series

mathematical analysis > calculus > series > convergent series

Preferred term

convergent series

Definition(s)

In mathematics, a series is the sum of the terms of an infinite sequence of numbers. More precisely, an infinite sequence ${\\displaystyle (a_{0},a_{1},a_{2},\\ldots )}$ defines a series $S$ that is denoted

${\\displaystyle S=a_{0}+a_{1}+a_{2}+\\cdots =\\sum _{k=0}^{\\infty }a_{k}.}$

The $n$ th partial sum $S n$ is the sum of the first $n$ terms of the sequence; that is,

${\\displaystyle S_{n}=\\sum _{k=1}^{n}a_{k}.}$

A series is convergent (or converges) if the sequence ${\\displaystyle (S_{1},S_{2},S_{3},\\dots )}$ of its partial sums tends to a limit; that means that, when adding one ${\\displaystyle a_{k}}$ after the other in the order given by the indices, one gets partial sums that become closer and closer to a given number. More precisely, a series converges, if there exists a number ${\\displaystyle \\ell }$ such that for every arbitrarily small positive number ${\\displaystyle \\varepsilon }$ , there is a (sufficiently large) integer ${\\displaystyle N}$ such that for all ${\\displaystyle n\\geq N}$ ,

${\\displaystyle \\left|S_{n}-\\ell \ ight|<\\varepsilon .}$

If the series is convergent, the (necessarily unique) number ${\\displaystyle \\ell }$ is called the sum of the series.
The same notation

${\\displaystyle \\sum _{k=1}^{\\infty }a_{k}}$

is used for the series, and, if it is convergent, to its sum. This convention is similar to that which is used for addition: $a + b$ denotes the operation of adding $a$ and $b$ as well as the result of this addition, which is called the sum of $a$ and $b$ .
Any series that is not convergent is said to be divergent or to diverge.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Convergent_series)