@prefix psr: . @prefix skos: . @prefix dc: . @prefix xsd: . psr: a skos:ConceptScheme . psr:-ZKV383X1-T skos:exactMatch , ; skos:prefLabel "série convergente"@fr, "convergent series"@en ; skos:definition """En mathématiques, une série est dite convergente si la suite de ses sommes partielles a une limite dans l'espace considéré. Dans le cas contraire, elle est dite divergente. Pour des séries numériques, ou à valeurs dans un espace de Banach — c'est-à-dire un espace vectoriel normé complet —, il suffit de prouver la convergence absolue de la série pour montrer sa convergence, ce qui permet de se ramener à une série à termes réels positifs. Pour étudier ces dernières, il existe une large variété de résultats, tous fondés sur le principe de comparaison.
(Wikipedia, L'Encylopédie Libre, https://fr.wikipedia.org/wiki/S%C3%A9rie_convergente)"""@fr, """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)"""@en ; skos:inScheme psr: ; dc:created "2023-08-03"^^xsd:date ; dc:modified "2023-08-03"^^xsd:date ; a skos:Concept ; skos:broader psr:-B3GGSQMX-3 . psr:-B3GGSQMX-3 skos:prefLabel "série"@fr, "series"@en ; a skos:Concept ; skos:narrower psr:-ZKV383X1-T .