## Series (mathematics) |

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In **series** is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity.^{[1]} The study of series is a major part of

For a long time, the idea that such a *never* reach the tortoise, and thus that movement does not exist. Zeno divided the race into infinitely many sub-races, each requiring a finite amount of time, so that the total time for Achilles to catch the tortoise is given by a series. The resolution of the paradox is that, although the series has an infinite number of terms, it has a finite sum, which gives the time necessary for Achilles to catch the tortoise.

In modern terminology, any (ordered) **infinite series**. Such a series is represented (or denoted) by an

or, using the

The infinite sequence of additions implied by a series cannot be effectively carried on (at least in a finite amount of time). However, if the set to which the terms and their finite sums belong has a notion of *sum* of the series. This value is the limit as *n* tends to infinity (if the limit exists) of the finite sums of the *n* first terms of the series, which are called the *n**th partial sums* of the series. That is,

When this limit exists, one says that the series is *convergent* or *summable*, or that the sequence is **summable**. In this case, the limit is called the *sum of the series*. Otherwise, the series is said to be *divergent*.

Generally, the terms of a series come from a

- basic properties
- examples of numerical series
- calculus and partial summation as an operation on sequences
- properties of series
- convergence tests
- series of functions
- history of the theory of infinite series
- generalizations
- see also
- notes
- references
- external links

An infinite series or simply a series is an infinite sum, represented by an ^{[2]}

where is any ordered

- .

If an abelian group *A* of terms has a concept of *A*, called the *sum of the series*. This includes the common cases from calculus in which the group is the field of real numbers or the field of complex numbers. Given a series its *k*th **partial sum** is

By definition, the series *converges* to the limit *L* (or simply *sums* to *L*), if the sequence of its partial sums has a limit *L*.^{[2]} In this case, one usually writes

A series is said to be *convergent* if it converges to some limit or *divergent* when it does not. The value of this limit, if it exists, is then the value of the series.

A series ∑*a*_{n} is said to *be convergent* when the sequence (*s*_{k}) of partial sums has a finite *s*_{k} is infinite or does not exist, the series is said to ^{[3]} When the limit of partial sums exists, it is called the value (or sum) of the series

An easy way that an infinite series can converge is if all the *a*_{n} are zero for *n* sufficiently large. Such a series can be identified with a finite sum, so it is only infinite in a trivial sense.

Working out the properties of the series that converge even if infinitely many terms are non-zero is the essence of the study of series. Consider the example

It is possible to "visualize" its convergence on the *equal* to 2 (although it is), but it does prove that it is *at most* 2. In other words, the series has an upper bound. Given that the series converges, proving that it is equal to 2 requires only elementary algebra. If the series is denoted *S*, it can be seen that

Therefore,

Mathematicians extend the idiom discussed earlier to other, equivalent notions of series. For instance, when we talk about a

we are talking, in fact, just about the series

But since these series always converge to ^{1}/_{9}. This leads to an argument that 9 × 0.111… = 0.999… = 1, which only relies on the fact that the limit laws for series preserve the arithmetic operations; this argument is presented in the article

Other Languages

العربية: متسلسلة (رياضيات)

asturianu: Serie matemática

azərbaycanca: Sıra (riyaziyyat)

Bân-lâm-gú: Kip-sò͘

башҡортса: Һанлы рәт

беларуская: Рад (матэматыка)

беларуская (тарашкевіца): Лікавы шэраг

български: Числов ред

bosanski: Red (matematika)

català: Sèrie (matemàtiques)

čeština: Řada (matematika)

dansk: Række (matematik)

Deutsch: Reihe (Mathematik)

Ελληνικά: Σειρά

español: Serie matemática

Esperanto: Serio (matematiko)

euskara: Serie (matematika)

فارسی: سری (ریاضیات)

français: Série (mathématiques)

galego: Serie (matemáticas)

贛語: 級數

客家語/Hak-kâ-ngî: Kip-sú

한국어: 급수 (수학)

हिन्दी: श्रेणी (गणित)

hrvatski: Red (matematika)

Bahasa Indonesia: Deret (matematika)

íslenska: Röð (stærðfræði)

italiano: Serie

עברית: טור (מתמטיקה)

ಕನ್ನಡ: ಶ್ರೇಢಿಗಳು (ಗಣಿತ)

ქართული: მწკრივი (მათემატიკა)

ລາວ: ຊຸດຈຳນວນ

Latina: Series (mathematica)

latviešu: Rinda (matemātika)

Lëtzebuergesch: Rei (Mathematik)

lietuvių: Skaičių eilutės

magyar: Numerikus sorok

македонски: Ред (математика)

മലയാളം: ശ്രേണി

Bahasa Melayu: Siri (matematik)

Nederlands: Reeks (wiskunde)

नेपाली: श्रेणी

日本語: 級数

norsk: Rekke (matematikk)

oʻzbekcha/ўзбекча: Qatorlar

Patois: Siiriz (matimatix)

polski: Szereg (matematyka)

português: Série (matemática)

română: Serie (matematică)

русский: Числовой ряд

Scots: Series (mathematics)

sicilianu: Seri (matimatica)

සිංහල: අපරිමිත ශ්රේණි

Simple English: Series

slovenčina: Rad (matematika)

slovenščina: Vrsta (matematika)

کوردی: زنجیرە (بیرکاری)

српски / srpski: Ред (математика)

srpskohrvatski / српскохрватски: Red (matematika)

suomi: Sarja (matematiikka)

svenska: Serie (matematik)

தமிழ்: தொடர் (கணிதம்)

ไทย: อนุกรม

Türkçe: Seri

українська: Ряд (математика)

اردو: سلسلہ (ریاضی)

Tiếng Việt: Chuỗi (toán học)

粵語: 級數

中文: 级数