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[[Berkas:Geometric Segment.svg|jmpl|Deret <math>\frac{1}{2}+\frac{1}{4}4+\frac{1}{8}+\cdots</math> konvergen menuju 1]]
'''Deret''' ({{lang-en|series}}) adalah [[Penambahan|jumlah]] suku-suku dari suatu [[barisan]]. Barisan dan deret hingga mempunyai elemen pertama dan terakhir yang terdefinisi, sedangkan barisan dan deret tak terhingga berlangsung terus menerus tak terbatas.<ref>p 264 '''[[Jan Gullberg|Gullberg, Jan]]:''', ''Mathematics: from the birth of numbers,'' W.W. Norton, 1997, hlm. 264, ISBN 0-393-04002-X</ref>
Dalam [[matematika]], jika ada suatu barisan bilangan [[Himpunan takhingga|tak hingga]] <math>\{a_n\}</math>, maka suatu deret secara mudahnya adalah hasil dari penambahan semua elemen-elemen itu bersama-sama: <math>a_1 + a_2 + a_3 + \cdots</math>. Ini dapat ditulis lebih ringkas menggunakan [[notasi Sigma]] ∑. Contohnya adalah deret terkenal dari [[Paradoks Zeno]] dan [[1/2 + 1/4 + 1/8 + 1/16 + ⋯|representasi matematikanya]]:
== Notasi ==
Simbol pada deret yaitu <math>\sum</math> menunjukkan penjumlahan dan dapat diinterpretasikan dengan mengulang hasil keliling (biasanya ditentukan di bawah penjumlahan), karena kita membutuhkan (biasanya [[bilangan bulat]]) nilai dalam rentang yang ditentukan (dari nilai awal ke batas atas), kemudian menambahkan ekspresi yang dihasilkan. Misalkan:
:<math>\sum_{k = 1}^{200} f(k) = f(1) + f(2) + \dots + f(200).</math>
:<math>\sum_{k = 1}^\infty a_k = \lim_{n \to \infty} \sum_{k = 1}^n a_k.</math>
Jika hasilnya limit tidak ada, deret tersebut dikatakan sebagai menyimpangdivergen.
Suatu deret dikatakan konvergen secara absolut jika deret yang terbentuk dari [[nilai absolut]] syarat pada konvergen; yaitu, diberi urutan tak terbatas <math>\{a_k\}</math>:
:<math>\sum_{k = 1}^\infty |a_k|</math>
konvergensi.[[Berkas:Exp series.gif|ka|jmpl|[[:en:exponential function|Fungsi eksponensial]] (biru), dan jumlah ''n''+1 elemen pertama dari [[:en:Maclaurin series|deret pangkat Maclaurin]] (merah).]]
== Deret Fourier ==
{{Lihat|Deret Fourier}}
== Sifat dasar ==
=== Definisi ===
Untuk setiap [[barisan]] <math>\{a_n\}</math> [[bilangan rasional]], [[bilangan real]], [[bilangan kompleks]], [[Fungsi (matematika)|fungsi]], dan lain-lain, '''deret''' yang bersangkutan didefinisikan sebagai [[jumlah formal]] tertata
:<math>\sum_{n=0}^{\infty}a_n = a_0 + a_1 + a_2 + \cdots </math>.
{{anchor|Jumlah parsial}}'''Barisan jumlah parsial''' <math>\{S_k\}</math> bersangkutan dengan suatu deret <math display="inline">\sum_{n=0}^\infty a_n</math> didefinisikan bagi setiap <math>k</math> sebagai jumlah Barisan <math>\{a_n\}</math> dari <math>a_0</math> sampaihingga <math>a_k</math>
:<math>S_k = \sum_{n=0}^{k}a_n = a_0 + a_1 + \cdots + a_k.</math>
Berdasarkan definisi, deret <math display="inline">\sum_{n=0}^{\infty} a_n</math> '''konvergen''' menjadi suatu limit <math>L</math> jika dan hanya jika urutan yang bersangkutan dengan jumlah parsial <math>\{S_k\}</math> [[Limit barisan#Definisi formal|convergeskonvegen]] menjadike <math>L</math>. Definisi ini biasanya ditulis sebagai
:<math>L = \sum_{n=0}^{\infty}a_n \Leftrightarrow L = \lim_{k \rightarrow \infty} S_k.</math>
== Deret fungsi ==
{{Main|Deret fungsi}}
:<math>f(x) = \sum_{n=0}^\infty a_n \left( x-c \right)^n = a_0 + a_1 (x-c)^1 + a_2 (x-c)^2 + a_3 (x-c)^3 + \cdots</math>
dengan ''a<submath>n a_n </submath>'' melambangkan koefisien suku ke-''<math> n'' </math>, ''<math> c'' </math> adalah konstanta dan ''<math> x'' </math> berubah-ubah di sekitar ''<math> c'' </math> (karena alasan ini, kadang-kadang deret seperti ini dikatakan ''berpusat'' di ''<math> c'' </math>). Deret ini biasanya berupa [[deret Taylor]] dari suatu [[fungsi]].
Pada banyak keadaan ''<math> c'' </math> sama dengan nol, contohnya pada [[:en:Maclaurin series|deret Maclaurin]]. Dalam hal tersebut [[deret pangkat]] mengambil bentuk yang lebih sederhana:<math>
: <math>f(x) = \sum_{n=0}^\infty a_n x^n = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \ldots</math>.
</math>
Deret pangkat biasanya ditemukan dalam [[analisis matematika]], tetapi juga dapat ditemukan pada [[kombinatorika]] (dengan nama [[fungsi pembangkit]]), dan pada [[teknik elektro]] (dengan nama [[transformasi Z]]).
[[Deret Taylor]] pada suatu titik ''c'' pada suatu fungsi adalah suatu deret pangkat yang dalam banyak kasus berkonvergen menjadi suatu fungsi dalam lingkungan ''<math> c </math>''. Misalnya, deret
:<math>\sum_{n=0}^{\infty} \frac{x^n}{n!}</math>
adalah deret Taylor dari <math>e^x</math> pada titik origin dan berkonvergen kepadanya untuk setiap ''<math>x''</math>.
<!--
Unless it converges only at ''x''=''c'', such a series converges on a certain open disc of convergence centered at the point ''c'' in the complex plane, and may also converge at some of the points of the boundary of the disc. The radius of this disc is known as the [[radius of convergence]], and can in principle be determined from the asymptotics of the coefficients ''a''<sub>''n''</sub>. The convergence is uniform on [[closed set|closed]] and [[bounded set|bounded]] (that is, [[compact set|compact]]) subsets of the interior of the disc of convergence: to wit, it is [[Compact convergence|uniformly convergent on compact sets]].
Historically, mathematicians such as [[Leonhard Euler]] operated liberally with infinite series, even if they were not convergent.
When calculus was put on a sound and correct foundation in the nineteenth century, rigorous proofs of the convergence of series were always required.
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|Deret Laurent}}
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|Deret Dirichlet}}
[[Deret Dirichlet]] mempunyai bentuk
:<math>\sum_{n=1}^\infty {a_n \over n^s},</math>
di mana ''s'' adalah suatu [[bilangan kompleks]]. For example, if all ''a''<sub>''n''</sub> are equal to 1, then the Dirichlet series is the [[Riemann zeta function]]
:<math>\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}.</math>
Like the zeta function, Dirichlet series in general play an important role in [[analytic number theory]]. Generally a Dirichlet series converges if the real part of ''s'' is greater than a number called the abscissa of convergence. In many cases, a Dirichlet series can be extended to an [[analytic function]] outside the domain of convergence by [[analytic continuation]]. For example, the Dirichlet series for the zeta function converges absolutely when Re ''s'' > 1, but the zeta function can be extended to a holomorphic function defined on <math>\mathbf{C}\setminus\{1\}</math> with a simple [[pole (complex analysis)|pole]] at 1.
This series can be directly generalized to [[general Dirichlet series]].
=== Deret trigonometri ===
{{Main|Deret trigonometri}}
A series of functions in which the terms are [[trigonometric function]]s is disebut '''Deret trigonometri''':
:<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.
==History of the theory of infinite series==
===Development of infinite series===
[[Greek mathematics|Greek]] mathematician [[Archimedes]] produced the first known summation of an infinite series with a
method that is still used in the area of calculus today. He used the [[method of exhaustion]] to calculate the [[area]] under the arc of a [[parabola]] with the summation of an infinite series, and gave a remarkably accurate approximation of [[Pi|π]].<ref>{{cite web | title = A history of calculus |author=O'Connor, J.J. and Robertson, E.F. | publisher = [[University of St Andrews]]| url = http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/The_rise_of_calculus.html |date=February 1996|accessdate= 2007-08-07}}</ref><ref>[http://eric.ed.gov/ERICWebPortal/custom/portlets/recordDetails/detailmini.jsp?_nfpb=true&_&ERICExtSearch_SearchValue_0=EJ502088&ERICExtSearch_SearchType_0=no&accno=EJ502088 Archimedes and Pi-Revisited.]</ref>
In the 17th century, [[James Gregory (astronomer and mathematician)|James Gregory]] worked in the new [[decimal]] system on infinite series and published several [[Maclaurin series]]. In 1715, a general method for constructing the [[Taylor series]] for all functions for which they exist was provided by [[Brook Taylor]]. [[Leonhard Euler]] in the 18th century, developed the theory of [[hypergeometric series]] and [[q-series]].
===Convergence criteria===
The investigation of the validity of infinite series is considered to begin with [[Carl Friedrich Gauss|Gauss]] in the 19th century. Euler had already considered the hypergeometric series
:<math>1 + \frac{\alpha\beta}{1\cdot\gamma}x + \frac{\alpha(\alpha+1)\beta(\beta+1)}{1 \cdot 2 \cdot \gamma(\gamma+1)}x^2 + \cdots</math>
on which Gauss published a memoir in 1812. It established simpler criteria of convergence, and the questions of remainders and the range of convergence.
[[Cauchy]] (1821) insisted on strict tests of convergence; he showed that if two series are convergent their product is not necessarily so, and with him begins the discovery of effective criteria. The terms ''convergence'' and ''divergence'' had been introduced long before by [[James Gregory (astronomer and mathematician)|Gregory]] (1668). [[Leonhard Euler]] and [[Carl Friedrich Gauss|Gauss]] had given various criteria, and [[Colin Maclaurin]] had anticipated some of Cauchy's discoveries. Cauchy advanced the theory of [[power series]] by his expansion of a complex [[function (mathematics)|function]] in such a form.
[[Niels Henrik Abel|Abel]] (1826) in his memoir on the [[binomial series]]
:<math>1 + \frac{m}{1!}x + \frac{m(m-1)}{2!}x^2 + \cdots</math>
corrected certain of Cauchy's conclusions, and gave a completely
scientific summation of the series for complex values of <math>m</math> and <math>x</math>. He showed the necessity of considering the subject of continuity in questions of convergence.
Cauchy's methods led to special rather than general criteria, and
the same may be said of [[Joseph Ludwig Raabe|Raabe]] (1832), who made the first elaborate
investigation of the subject, of [[Augustus De Morgan|De Morgan]] (from 1842), whose
logarithmic test [[Paul du Bois-Reymond|DuBois-Reymond]] (1873) and [[Alfred Pringsheim|Pringsheim]] (1889) have
shown to fail within a certain region; of [[Joseph Louis François Bertrand|Bertrand]] (1842), [[Pierre Ossian Bonnet|Bonnet]]
(1843), [[Carl Johan Malmsten|Malmsten]] (1846, 1847, the latter without integration);
[[George Gabriel Stokes|Stokes]] (1847), [[Paucker]] (1852), [[Chebyshev]] (1852), and [[Arndt]]
(1853).
General criteria began with [[Ernst Kummer|Kummer]] (1835), and have been
studied by [[Gotthold Eisenstein|Eisenstein]] (1847), [[Weierstrass]] in his various
contributions to the theory of functions, [[Ulisse Dini|Dini]] (1867),
DuBois-Reymond (1873), and many others. Pringsheim's memoirs (1889) present the most complete general theory.
===Uniform convergence===
The theory of [[uniform convergence]] was treated by Cauchy (1821), his
limitations being pointed out by Abel, but the first to attack it
successfully were [[Philipp Ludwig von Seidel|Seidel]] and [[George Gabriel Stokes|Stokes]] (1847–48). Cauchy took up the
problem again (1853), acknowledging Abel's criticism, and reaching
the same conclusions which Stokes had already found. Thomae used the
doctrine (1866), but there was great delay in recognizing the
importance of distinguishing between uniform and non-uniform
convergence, in spite of the demands of the theory of functions.
===Semi-convergence===
A series is said to be semi-convergent (or conditionally convergent) if it is convergent but not [[absolute convergence|absolutely convergent]].
Semi-convergent series were studied by Poisson (1823), who also gave a general form for the remainder of the Maclaurin formula. The most important solution of the problem is due, however, to Jacobi (1834),
who attacked the question of the remainder from a different standpoint and reached a different formula. This expression was also worked out, and another one given, by [[Carl Johan Malmsten|Malmsten]] (1847). [[Schlömilch]] (''Zeitschrift'', Vol.I, p. 192, 1856) also improved Jacobi's remainder, and showed the relation between the remainder and [[Faulhaber's formula|Bernoulli's function]]
:<math>F(x) = 1^n + 2^n + \cdots + (x - 1)^n.\,</math>
[[Angelo Genocchi|Genocchi]] (1852) has further contributed to the theory.
Among the early writers was [[Josef Hoene-Wronski|Wronski]], whose "loi suprême" (1815) was hardly recognized until [[Arthur Cayley|Cayley]] (1873) brought it into
prominence.
=== Deret Fourier ===
[[Fourier series]] were being investigated
as the result of physical considerations at the same time that
Gauss, Abel, and Cauchy were working out the theory of infinite
series. Series for the expansion of sines and cosines, of multiple
arcs in powers of the sine and cosine of the arc had been treated by
[[Jacob Bernoulli]] (1702) and his brother [[Johann Bernoulli]] (1701) and still
earlier by [[Franciscus Vieta|Vieta]]. Euler and [[Joseph Louis Lagrange|Lagrange]] simplified the subject,
as did [[Louis Poinsot|Poinsot]], [[Karl Schröter|Schröter]], [[James Whitbread Lee Glaisher|Glaisher]], and [[Ernst Kummer|Kummer]].
Fourier (1807) set for himself a different problem, to
expand a given function of ''x'' in terms of the sines or cosines of
multiples of ''x'', a problem which he embodied in his ''[[Théorie analytique de la chaleur]]'' (1822). Euler had already given the
formulas for determining the coefficients in the series;
Fourier was the first to assert and attempt to prove the general
theorem. [[Siméon Denis Poisson|Poisson]] (1820–23) also attacked the problem from a
different standpoint. Fourier did not, however, settle the question
of convergence of his series, a matter left for [[Augustin Louis Cauchy|Cauchy]] (1826) to
attempt and for Dirichlet (1829) to handle in a thoroughly
scientific manner (see [[convergence of Fourier series]]). Dirichlet's treatment (''[[Crelle]]'', 1829), of trigonometric series was the subject of criticism and improvement by
Riemann (1854), Heine, [[Rudolf Lipschitz|Lipschitz]], [[Ludwig Schläfli|Schläfli]], and
[[Paul du Bois-Reymond|du Bois-Reymond]]. Among other prominent contributors to the theory of
trigonometric and Fourier series were [[Ulisse Dini|Dini]], [[Charles Hermite|Hermite]], [[Georges Henri Halphen|Halphen]],
Krause, Byerly and [[Paul Émile Appell|Appell]].
==Generalisasi==
=== Deret Asimptotik ===
[[Asymptotic series]], otherwise [[asymptotic expansion]]s, are infinite series whose partial sums become good approximations in the limit of some point of the domain. In general they do not converge. But they are useful as sequences of approximations, each of which provides a value close to the desired answer for a finite number of terms. The difference is that an asymptotic series cannot be made to produce an answer as exact as desired, the way that convergent series can. In fact, after a certain number of terms, a typical [[asymptotic series]] reaches its best approximation; if more terms are included, most such series will produce worse answers.
=== Deret Divergen ===
{{Main|Divergent series}}
Under many circumstances, it is desirable to assign a limit to a series which fails to converge in the usual sense. A [[summability method]] is such an assignment of a limit to a subset of the set of divergent series which properly extends the classical notion of convergence. Summability methods include [[Cesàro summation]], (''C'',''k'') summation, [[Abel summation]], and [[Borel summation]], in increasing order of generality (and hence applicable to increasingly divergent series).
A variety of general results concerning possible summability methods are known. The [[Silverman–Toeplitz theorem]] characterizes ''matrix summability methods'', which are methods for summing a divergent series by applying an infinite matrix to the vector of coefficients. The most general method for summing a divergent series is non-constructive, and concerns [[Banach limit]]s.
===Deret dalam ruang Banach ===
The notion of series can be easily extended to the case of a [[Banach space]]. If ''x''<sub>''n''</sub> is a sequence of elements of a Banach space ''X'', then the series Σ''x''<sub>''n''</sub> converges to ''x'' ∈ ''X'' if the sequence of partial sums of the series tends to ''x''; to wit,
:<math>\biggl\|x - \sum_{n=0}^N x_n\biggr\|\to 0</math>
as ''N'' → ∞.
More generally, convergence of series can be defined in any [[abelian group|abelian]] [[Hausdorff space|Hausdorff]] [[topological group]]. Specifically, in this case, Σ''x''<sub>''n''</sub> converges to ''x'' if the sequence of partial sums converges to ''x''.
===Summasi atas himpunan indeks sembarang ===
Definitions may be given for sums over an arbitrary index set ''I''. There are two main differences with the usual notion of series: first, there is no specific order given on the set ''I''; second, this set ''I'' may be uncountable.
====Famili bilangan non-negatif====
When summing a family {''a''<sub>''i''</sub>}, ''i'' ∈ ''I'', of non-negative numbers, one may define
:<math>\sum_{i\in I}a_i = \sup \Bigl\{ \sum_{i\in A}a_i\,\big| A \text{ finite, } A \subset I\Bigr\} \in [0, +\infty].</math>
When the sum is finite, the set of ''i'' ∈ ''I'' such that ''a<sub>i</sub>'' > 0 is countable. Indeed for every ''n'' ≥ 1, the set <math>\scriptstyle A_n = \{ i \in I \,:\, a_i > 1/n \}</math> is finite, because
:<math> \frac 1 n \, \textrm{card}(A_n) \le \sum_{i\in A_n} a_i \le \sum_{i\in I}a_i < \infty.</math>
If ''I'' is countably infinite and enumerated as ''I'' = {''i''<sub>0</sub>, ''i''<sub>1</sub>,...} then the above defined sum satisfies
:<math>\sum_{i \in I} a_i = \sum_{k=0}^{+\infty} a_{i_k},</math>
provided the value ∞ is allowed for the sum of the series.
Any sum over non-negative reals can be understood as the integral of a non-negative function with respect to the [[counting measure]], which accounts for the many similarities between the two constructions.
==== Grup topologi Abel ====
Let ''a'' : ''I'' → ''X'', where ''I'' is any set and ''X'' is an [[abelian group|abelian]] [[Hausdorff space|Hausdorff]] [[topological group]]. Let ''F'' be the collection of all [[finite set|finite]] [[subset]]s of ''I''. Note that ''F'' is a [[directed set]] [[Partially ordered set|ordered]] under [[inclusion (mathematics)|inclusion]] with [[union (set theory)|union]] as [[join (mathematics)|join]]. Define the sum ''S'' of the family ''a'' as the limit
:<math> S = \sum_{i\in I}a_i = \lim \Bigl\{\sum_{i\in A}a_i\,\big| A\in F\Bigr\}</math>
if it exists and say that the family ''a'' is unconditionally summable. Saying that the sum ''S'' is the limit of finite partial sums means that for every neighborhood ''V'' of 0 in ''X'', there is a finite subset ''A''<sub>0</sub> of ''I'' such that
:<math>S - \sum_{i \in A} a_i \in V, \quad A \supset A_0.</math>
Because ''F'' is not [[total order|totally ordered]], this is not a [[limit of a sequence]] of partial sums, but rather of a [[net (mathematics)|net]].<ref name="Bourbaki">{{cite book|title=General Topology: Chapters 1-4 |first=Nicolas |last=Bourbaki |authorlink=Nicolas Bourbaki |year=1998 |publisher=Springer |isbn=9783540642411 |pages=261–270}}</ref><ref name="Choquet">{{cite book|title=Topology |url=https://archive.org/details/topology00choq |first=Gustave |last=Choquet |authorlink=Gustave Choquet |year=1966 |publisher=Academic Press |isbn=9780121734503 |pages=[https://archive.org/details/topology00choq/page/216 216]–231}}</ref>
For every ''W'', neighborhood of 0 in ''X'', there is a smaller neighborhood ''V'' such that ''V'' − ''V'' ⊂ ''W''. It follows that the finite partial sums of an unconditionally summable family ''a<sub>i</sub>'', ''i'' ∈ ''I'', form a ''Cauchy net'', that is: for every ''W'', neighborhood of 0 in ''X'', there is a finite subset ''A''<sub>0</sub> of ''I'' such that
:<math>\sum_{i \in A_1} a_i - \sum_{i \in A_2} a_i \in W, \quad A_1, A_2 \supset A_0.</math>
When ''X'' is [[Complete metric space|complete]], a family ''a'' is unconditionally summable in ''X'' if and only if the finite sums satisfy the latter Cauchy net condition. When ''X'' is complete and ''a<sub>i</sub>'', ''i'' ∈ ''I'', is unconditionally summable in ''X'', then for every subset ''J'' ⊂ ''I'', the corresponding subfamily ''a<sub>j</sub>'', ''j'' ∈ ''J'', is also unconditionally summable in ''X''.
When the sum of a family of non-negative numbers, in the extended sense defined before, is finite, then it coincides with the sum in the topological group ''X'' = '''R'''.
If a family ''a'' in ''X'' is unconditionally summable, then for every ''W'', neighborhood of 0 in ''X'', there is a finite subset ''A''<sub>0</sub> of ''I'' such that ''a''<sub>''i''</sub> ∈ ''W'' for every ''i'' not in ''A''<sub>0</sub>. If ''X'' is [[first-countable space|first-countable]], it follows that the set of ''i'' ∈ ''I'' such that ''a<sub>i</sub>'' ≠ 0 is countable. This need not be true in a general abelian topological group (see examples below).
==== Deret konvergen tak bersyarat ====
Suppose that ''I'' = '''N'''. If a family ''a''<sub>''n''</sub>, ''n'' ∈ '''N''', is unconditionally summable in an abelian Hausdorff topological group ''X'', then the series in the usual sense converges and has the same sum,
:<math>\sum_{n=0}^\infty a_n = \sum_{n \in \mathbf{N}} a_n.</math>
By nature, the definition of unconditional summability is insensitive to the order of the summation. When ∑''a''<sub>''n''</sub> is unconditionally summable, then the series remains convergent after any permutation ''σ'' of the set '''N''' of indices, with the same sum,
:<math>\sum_{n=0}^\infty a_{\sigma(n)} = \sum_{n=0}^\infty a_n.</math>
Conversely, if every permutation of a series ∑''a''<sub>''n''</sub> converges, then the series is unconditionally convergent. When ''X'' is complete, then unconditional convergence is also equivalent to the fact that all subseries are convergent; if ''X'' is a Banach space, this is equivalent to say that for every sequence of signs ''ε''<sub>''n''</sub> = 1 or −1, the series
:<math>\sum_{n=0}^\infty \varepsilon_n a_n</math>
converges in ''X''. If ''X'' is a Banach space, then one may define the notion of absolute convergence. A series ∑''a''<sub>''n''</sub> of vectors in ''X'' converges absolutely if
:<math> \sum_{n \in \mathbf{N}} \|a_n\| < +\infty.</math>
If a series of vectors in a Banach space converges absolutely then it converges unconditionally, but the converse only holds in finite-dimensional Banach spaces (theorem of {{harvtxt|Dvoretzky|Rogers|1950}}).
==== Jumlah tertata baik ====
Conditionally convergent series can be considered if ''I'' is a [[well-ordered]] set, for example an [[ordinal number]] ''α''<sub>0</sub>. One may define by [[transfinite recursion]]:
:<math>\sum_{\beta < \alpha + 1} a_\beta = a_{\alpha} + \sum_{\beta < \alpha} a_\beta\,\!</math>
and for a limit ordinal ''α'',
:<math>\sum_{\beta < \alpha} a_\beta = \lim_{\gamma\to\alpha} \sum_{\beta < \gamma} a_\beta</math>
if this limit exists. If all limits exist up to ''α''<sub>0</sub>, then the series converges.
==== Contoh ====
{{ordered list
|1= Given a function ''f'' : ''X''→''Y'', with ''Y'' an abelian topological group, define for every ''a'' ∈ ''X''
:<math>f_a(x)=
\begin{cases}
0 & x\neq a, \\
f(a) & x=a, \\
\end{cases}
</math>
a function whose [[support (mathematics)|support]] is a [[Singleton (mathematics)|singleton]] {''a''}. Then
:<math>f=\sum_{a \in X}f_a</math>
in the [[topology of pointwise convergence]] (that is, the sum is taken in the infinite product group ''Y''<sup>''X'' </sup>).
|2= In the definition of [[partitions of unity]], one constructs sums of functions over arbitrary index set ''I'',
:<math> \sum_{i \in I} \varphi_i(x) = 1.</math>
While, formally, this requires a notion of sums of uncountable series, by construction there are, for every given ''x'', only finitely many nonzero terms in the sum, so issues regarding convergence of such sums do not arise. Actually, one usually assumes more: the family of functions is ''locally finite'', ''i.e.'', for every ''x'' there is a neighborhood of ''x'' in which all but a finite number of functions vanish. Any regularity property of the ''φ<sub>i</sub>'', such as continuity, differentiability, that is preserved under finite sums will be preserved for the sum of any subcollection of this family of functions.
|3= On the [[first uncountable ordinal]] ω<sub>1</sub> viewed as a topological space in the [[order topology]], the constant function ''f'': [0,ω<sub>1</sub>) → [0,ω<sub>1</sub>] given by ''f''(α) = 1 satisfies
:<math>\sum_{\alpha\in[0,\omega_1)}f(\alpha) = \omega_1</math>
(in other words, ω<sub>1</sub> copies of 1 is ω<sub>1</sub>) only if one takes a limit over all ''countable'' partial sums, rather than finite partial sums. This space is not separable.
}}
-->:
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