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Istilah-istilah dalam suatu deret sering diproduksi menurut kaidah tertentu, misalnya dengan suatu [[rumus]], atau melalui suatu [[algoritme]]. Mengingat tidak terbatasnya jumlah istilah, hasilnya sering disebut '''deret tak terhingga''' (''infinite series''). Berbeda dengan finite summations, deret tak terhingga membutuhkan bantuan dari [[analisis matematika]], dan secara khusus [[limit (matematika)|limit]], untuk dapat dipahami dan dimanipulasi secara penuh. Selain jumlahnya yang banyak dalam matematika, deret tak terhingga juga sering digunakan dalam bidang-bidang kuantitatif lain seperti [[fisika]], sains komputer, dan finansial.
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===Definisi===
Untuk setiap [[urutan]] <math>\{a_n\}</math> [[bilangan rasional]], [[bilangan real]], [[bilangan kompleks]], [[Fungsi (matematika)|fungsi]], dan lain-lain, '''deret''' yang bersangkutan didefinisikan sebagai [[formal sum]] tertata
:<math>\sum_{n=0}^{\infty}a_n = a_0 + a_1 + a_2 + \cdots </math>.
:<math>S_k = \sum_{n=0}^{k}a_n = a_0 + a_1 + \cdots + a_k.</math>
:<math>L = \sum_{n=0}^{\infty}a_n \Leftrightarrow L = \lim_{k \rightarrow \infty} S_k.</math>
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More generally, if <math>I \xrightarrow{a} G</math> is a [[Function (mathematics)|function]] from an [[index set]] I to a set G, then the '''series''' associated to <math>a</math> is the [[formal sum]] of the elements <math>a(x) \in G </math> over the index elements <math>x \in I</math> denoted by the
:<math>\sum_{x \in I} a(x).</math>
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*[[Ratio test]]: If there exists a constant ''C'' < 1 such that |''a''<sub>''n''+1</sub>/''a''<sub>''n''</sub>|<''C'' for all sufficiently large ''n'', then ∑''a''<sub>''n''</sub> converges absolutely. When the ratio is less than 1, but not less than a constant less than 1, convergence is possible but this test does not establish it.
*[[Root test]]: If there exists a constant ''C'' < 1 such that |''a''<sub>''n''</sub>|<sup>1/''n''</sup> ≤ ''C'' for all sufficiently large ''n'', then ∑''a''<sub>''n''</sub> converges absolutely.
*[[Integral test for convergence|Integral test]]: if ''ƒ''(''x'') is a positive [[monotone decreasing]] function defined on the [[interval (mathematics)|interval]] <nowiki>[</nowiki>1, ∞<nowiki>)</nowiki>--><!--DO NOT "FIX" THE "TYPO" IN THE FOREGOING. IT IS INTENDED TO SAY [...) WITH A SQUARE BRACKET ON THE LEFT AND A ROUND BRACKET ON THE RIGHT. --><!-- with ''ƒ''(''n'') = ''a''<sub>''n''</sub> for all ''n'', then ∑''a''<sub>''n''</sub> converges if and only if the [[integral]]  ∫<sub>1</sub><sup>∞</sup> ''ƒ''(''x'') d''x'' is finite.
*[[Cauchy's condensation test]]: If ''a''<sub>''n''</sub> is non-negative and non-increasing, then the two series  ∑''a''<sub>''n''</sub>  and  ∑2<sup>''k''</sup>''a''<sub>(2<sup>''k''</sup>)</sub> are of the same nature: both convergent, or both divergent.
*[[Alternating series test]]: A series of the form ∑(−1)<sup>''n''</sup> ''a''<sub>''n''</sub> (with ''a''<sub>''n''</sub> ≥ 0) is called ''alternating''. Such a series converges if the [[sequence]] ''a''<sub>''n''</sub> is [[monotone decreasing]] and converges to 0. The converse is in general not true.
*For some specific types of series there are more specialized convergence tests, for instance for [[Fourier series]] there is the [[Dini test]].
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{{Main|
Suatu deret fungsi-fungsi bernilai real atau kompleks
:<math>\sum_{n=0}^\infty f_n(x)</math>
'''[[Pointwise convergence|converges pointwise]]'''
:<math>s_N(x) = \sum_{n=0}^N f_n(x)</math>
converge
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A stronger notion of convergence of a series of functions is called '''[[uniform convergence]]'''. The series converges uniformly if it converges pointwise to the function ''ƒ''(''x''), and the error in approximating the limit by the ''N''th partial sum,
:<math>|s_N(x) - f(x)|\ </math>
can be made minimal ''independently'' of ''x'' by choosing a sufficiently large ''N''.
Uniform convergence is desirable for a series because many properties of the terms of the series are then retained by the limit. For example, if a series of continuous functions converges uniformly, then the limit function is also continuous. Similarly, if the ''ƒ''<sub>''n''</sub> are [[integral|integrable]] on a closed and bounded interval ''I'' and converge uniformly, then the series is also integrable on ''I'' and can be integrated term-by-term. Tests for uniform convergence include the [[Weierstrass M-test|Weierstrass' M-test]], [[Abel's uniform convergence test]], [[Dini's test]]--><!-- , and the [[Cauchy criterion]]: this is not about convergence of functions, even less about uniform convergence. -->.
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More sophisticated types of convergence of a series of functions can also be defined. In [[measure theory]], for instance, a series of functions converges [[almost everywhere]] if it converges pointwise except on a certain set of [[null set|measure zero]]. Other [[modes of convergence]] depend on a different [[metric space]] structure on the space of functions under consideration. For instance, a series of functions '''converges in mean''' on a set ''E'' to a limit function ''ƒ'' provided
:<math>\int_E \left|s_N(x)-f(x)\right|^2\,dx \to 0</math>
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However, the formal operation with non-convergent series has been retained in rings of [[formal power series]] which are studied in [[abstract algebra]]. Formal power series are also used in [[combinatorics]] to describe and study [[sequence]]s that are otherwise difficult to handle; this is the method of [[generating function]]s.
=== Deret Laurent
{{Main|
Laurent series generalize power series by admitting terms into the series with negative as well as positive exponents. A Laurent series is thus any series of the form
:<math>\sum_{n=-\infty}^\infty a_n x^n.</math>
If such a series converges, then in general it does so in an [[annulus (mathematics)|annulus]] rather than a disc, and possibly some boundary points. The series converges uniformly on compact subsets of the interior of the annulus of convergence.
===Deret Dirichlet
:{{Main|
:<math>\sum_{n=1}^\infty {a_n \over n^s},</math>
:<math>\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}.</math>
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This series can be directly generalized to [[general Dirichlet series]].
=== Deret trigonometri ===
{{Main|
A series of functions in which the terms are [[trigonometric function]]s is
:<math>\tfrac12 A_0 + \sum_{n=1}^\infty \left(A_n\cos nx + B_n \sin nx\right).</math>
The most important example of a trigonometric series is the [[Fourier series]] of a function.
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