Series (mathematics)

Illustration of 3 geometric series with partial sums from 1 to 6 terms. The dashed line represents the limit.

A series Σa_n is said to converge or to be convergent when the sequence (s_k) of partial sums has a finite limit. If the limit of s_k is infinite or does not exist, the series is said to diverge.[4][2] When the limit of partial sums exists, it is called the value (or sum) of the series

$${\displaystyle \sum _{n=0}^{\infty }a_{n}=\lim _{k\to \infty }s_{k}=\lim _{k\to \infty }\sum _{n=0}^{k}a_{n}.}$$

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 nonzero, is the essence of the study of series. Consider the example

$${\displaystyle 1+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+\cdots +{\frac {1}{2^{n}}}+\cdots .}$$

It is possible to "visualize" its convergence on the real number line: we can imagine a line of length 2, with successive segments marked off of lengths 1, 1/2, 1/4, etc. There is always room to mark the next segment, because the amount of line remaining is always the same as the last segment marked: When we have marked off 1/2, we still have a piece of length 1/2 unmarked, so we can certainly mark the next 1/4. This argument does not prove that the sum is 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

$${\displaystyle S/2={\frac {1+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+\cdots }{2}}={\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+\cdots .}$$

Therefore,

$${\displaystyle S-S/2=1\Rightarrow S=2.}$$

The idiom can be extended to other, equivalent notions of series. For instance, a recurring decimal, as in

$${\displaystyle x=0.111\dots ,}$$

encodes the series

$${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{10^{n}}}.}$$

Since these series always converge to real numbers (because of what is called the completeness property of the real numbers), to talk about the series in this way is the same as to talk about the numbers for which they stand. In particular, the decimal expansion 0.111... can be identified with 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; for more detail on this argument, see 0.999....

$${\displaystyle \sum _{i=1}^{\infty }{\frac {(-1)^{i+1}}{i}}=\ln 2}$$

[2]

$${\displaystyle \sum _{i=0}^{\infty }{\frac {1}{(2i+1)(2i+2)}}=\ln 2}$$

$${\displaystyle \sum _{i=0}^{\infty }{\frac {(-1)^{i}}{(i+1)(i+2)}}=2\ln(2)-1}$$

$${\displaystyle \sum _{i=1}^{\infty }{\frac {1}{i\left(4i^{2}-1\right)}}=2\ln(2)-1}$$

$${\displaystyle \sum _{i=1}^{\infty }{\frac {1}{2^{i}i}}=\ln 2}$$

$${\displaystyle \sum _{i=1}^{\infty }\left({\frac {1}{3^{i}}}+{\frac {1}{4^{i}}}\right){\frac {1}{i}}=\ln 2}$$

$${\displaystyle \sum _{i=1}^{\infty }{\frac {1}{2i(2i-1)}}=\ln 2}$$

When a_n is a non-negative real number for every n, the sequence S_N of partial sums is non-decreasing. It follows that a series Σa_n with non-negative terms converges if and only if the sequence S_N of partial sums is bounded.

For example, the series

$${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{2}}}}$$

is convergent, because the inequality

$${\displaystyle {\frac {1}{n^{2}}}\leq {\frac {1}{n-1}}-{\frac {1}{n}},\quad n\geq 2,}$$

and a telescopic sum argument implies that the partial sums are bounded by 2. The exact value of the original series is the Basel problem.

When you group a series reordering of the series does not happen, so Riemann series theorem does not apply. A new series will have its partial sums as subsequence of original series, which means if the original series converges, so does the new series. But for divergent series that is not true, for example 1-1+1-1+... grouped every two elements will create 0+0+0+... series, which is convergent. On the other hand, divergence of the new series means the original series can be only divergent which is sometimes useful, like in Oresme proof.

A series

$${\displaystyle \sum _{n=0}^{\infty }a_{n}}$$

converges absolutely if the series of absolute values

$${\displaystyle \sum _{n=0}^{\infty }\left|a_{n}\right|}$$

converges. This is sufficient to guarantee not only that the original series converges to a limit, but also that any reordering of it converges to the same limit.

A series of real or complex numbers is said to be conditionally convergent (or semi-convergent) if it is convergent but not absolutely convergent. A famous example is the alternating series

$${\displaystyle \sum \limits _{n=1}^{\infty }{(-1)^{n+1} \over n}=1-{1 \over 2}+{1 \over 3}-{1 \over 4}+{1 \over 5}-\cdots ,}$$

which is convergent (and its sum is equal to ${\displaystyle \ln 2}$), but the series formed by taking the absolute value of each term is the divergent harmonic series. The Riemann series theorem says that any conditionally convergent series can be reordered to make a divergent series, and moreover, if the ${\displaystyle a_{n}}$ are real and ${\displaystyle S}$ is any real number, that one can find a reordering so that the reordered series converges with sum equal to ${\displaystyle S}$.

Abel's test is an important tool for handling semi-convergent series. If a series has the form

$${\displaystyle \sum a_{n}=\sum \lambda _{n}b_{n}}$$

where the partial sums ${\displaystyle B_{n}=b_{0}+\cdots +b_{n}}$ are bounded, ${\displaystyle \lambda _{n}}$ has bounded variation, and ${\displaystyle \lim \lambda _{n}b_{n}}$ exists:

$${\displaystyle \sup _{N}\left|\sum _{n=0}^{N}b_{n}\right|<\infty ,\ \ \sum \left|\lambda _{n+1}-\lambda _{n}\right|<\infty \ {\text{and}}\ \lambda _{n}B_{n}\ {\text{converges,}}}$$

then the series ${\textstyle \sum a_{n}}$ is convergent. This applies to the point-wise convergence of many trigonometric series, as in

$${\displaystyle \sum _{n=2}^{\infty }{\frac {\sin(nx)}{\ln n}}}$$

with ${\displaystyle 0<x<2\pi }$. Abel's method consists in writing ${\displaystyle b_{n+1}=B_{n+1}-B_{n}}$, and in performing a transformation similar to integration by parts (called summation by parts), that relates the given series ${\textstyle \sum a_{n}}$ to the absolutely convergent series

$${\displaystyle \sum (\lambda _{n}-\lambda _{n+1})\,B_{n}.}$$

The evaluation of truncation errors is an important procedure in numerical analysis (especially validated numerics and computer-assisted proof).

When conditions of the alternating series test are satisfied by ${\textstyle S:=\sum _{m=0}^{\infty }(-1)^{m}u_{m}}$, there is an exact error evaluation.[7] Set ${\displaystyle s_{n}}$ to be the partial sum ${\textstyle s_{n}:=\sum _{m=0}^{n}(-1)^{m}u_{m}}$ of the given alternating series ${\displaystyle S}$. Then the next inequality holds:

$${\displaystyle |S-s_{n}|\leq u_{n+1}.}$$

Taylor's theorem is a statement that includes the evaluation of the error term when the Taylor series is truncated.

By using the ratio, we can obtain the evaluation of the error term when the hypergeometric series is truncated.[8]

For the matrix exponential:

$${\displaystyle \exp(X):=\sum _{k=0}^{\infty }{\frac {1}{k!}}X^{k},\quad X\in \mathbb {C} ^{n\times n},}$$

the following error evaluation holds (scaling and squaring method):[9][10][11]

$${\displaystyle T_{r,s}(X):=\left[\sum _{j=0}^{r}{\frac {1}{j!}}(X/s)^{j}\right]^{s},\quad \|\exp(X)-T_{r,s}(X)\|\leq {\frac {\|X\|^{r+1}}{s^{r}(r+1)!}}\exp(\|X\|).}$$

Main article: Power series

A power series is a series of the form

$${\displaystyle \sum _{n=0}^{\infty }a_{n}(x-c)^{n}.}$$

The Taylor series at a point c of a function is a power series that, in many cases, converges to the function in a neighborhood of c. For example, the series

$${\displaystyle \sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}}$$

is the Taylor series of ${\displaystyle e^{x}}$ at the origin and converges to it for every x.

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_n. The convergence is uniform on closed and bounded (that is, compact) subsets of the interior of the disc of convergence: to wit, it is 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.

While many uses of power series refer to their sums, it is also possible to treat power series as formal sums, meaning that no addition operations are actually performed, and the symbol "+" is an abstract symbol of conjunction which is not necessarily interpreted as corresponding to addition. In this setting, the sequence of coefficients itself is of interest, rather than the convergence of the series. Formal power series are used in combinatorics to describe and study sequences that are otherwise difficult to handle, for example, using the method of generating functions. The Hilbert–Poincaré series is a formal power series used to study graded algebras.

Even if the limit of the power series is not considered, if the terms support appropriate structure then it is possible to define operations such as addition, multiplication, derivative, antiderivative for power series "formally", treating the symbol "+" as if it corresponded to addition. In the most common setting, the terms come from a commutative ring, so that the formal power series can be added term-by-term and multiplied via the Cauchy product. In this case the algebra of formal power series is the total algebra of the monoid of natural numbers over the underlying term ring.[12] If the underlying term ring is a differential algebra, then the algebra of formal power series is also a differential algebra, with differentiation performed term-by-term.

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

$${\displaystyle \sum _{n=-\infty }^{\infty }a_{n}x^{n}.}$$

If such a series converges, then in general it does so in an 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.

Main article: Dirichlet series

A Dirichlet series is one of the form

$${\displaystyle \sum _{n=1}^{\infty }{a_{n} \over n^{s}},}$$

where s is a complex number. For example, if all a_n are equal to 1, then the Dirichlet series is the Riemann zeta function

$${\displaystyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}.}$$

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 ${\displaystyle \mathbb {C} \setminus \{1\}}$ with a simple pole at 1.

This series can be directly generalized to general Dirichlet series.

A series of functions in which the terms are trigonometric functions is called a trigonometric series:

$${\displaystyle {\frac {1}{2}}A_{0}+\sum _{n=1}^{\infty }\left(A_{n}\cos nx+B_{n}\sin nx\right).}$$

The most important example of a trigonometric series is the Fourier series of a function.

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 π.[13][14]

Mathematicians from the Kerala school were studying infinite series c. 1350 CE.[15]

In the 17th century, 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.

The investigation of the validity of infinite series is considered to begin with Gauss in the 19th century. Euler had already considered the hypergeometric series

$${\displaystyle 1+{\frac {\alpha \beta }{1\cdot \gamma }}x+{\frac {\alpha (\alpha +1)\beta (\beta +1)}{1\cdot 2\cdot \gamma (\gamma +1)}}x^{2}+\cdots }$$

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 Gregory (1668). Leonhard Euler and 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 in such a form.

Abel (1826) in his memoir on the binomial series

$${\displaystyle 1+{\frac {m}{1!}}x+{\frac {m(m-1)}{2!}}x^{2}+\cdots }$$

corrected certain of Cauchy's conclusions, and gave a completely scientific summation of the series for complex values of ${\displaystyle m}$ and ${\displaystyle x}$. 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 Raabe (1832), who made the first elaborate investigation of the subject, of De Morgan (from 1842), whose logarithmic test DuBois-Reymond (1873) and Pringsheim (1889) have shown to fail within a certain region; of Bertrand (1842), Bonnet (1843), Malmsten (1846, 1847, the latter without integration); Stokes (1847), Paucker (1852), Chebyshev (1852), and Arndt (1853).

General criteria began with Kummer (1835), and have been studied by Eisenstein (1847), Weierstrass in his various contributions to the theory of functions, Dini (1867), DuBois-Reymond (1873), and many others. Pringsheim's memoirs (1889) present the most complete general theory.

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 Seidel and 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.

A series is said to be semi-convergent (or conditionally convergent) if it is convergent but not 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 Malmsten (1847). Schlömilch (Zeitschrift, Vol.I, p. 192, 1856) also improved Jacobi's remainder, and showed the relation between the remainder and Bernoulli's function

$${\displaystyle F(x)=1^{n}+2^{n}+\cdots +(x-1)^{n}.}$$

Genocchi (1852) has further contributed to the theory.

Among the early writers was Wronski, whose "loi suprême" (1815) was hardly recognized until Cayley (1873) brought it into prominence.

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 Vieta. Euler and Lagrange simplified the subject, as did Poinsot, Schröter, Glaisher, and 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. 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 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, Lipschitz, Schläfli, and du Bois-Reymond. Among other prominent contributors to the theory of trigonometric and Fourier series were Dini, Hermite, Halphen, Krause, Byerly and Appell.

Asymptotic series, otherwise asymptotic expansions, 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.

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 limits.

Definitions may be given for sums over an arbitrary index set ${\displaystyle I.}$[16] There are two main differences with the usual notion of series: first, there is no specific order given on the set ${\displaystyle I}$; second, this set ${\displaystyle I}$ may be uncountable. The notion of convergence needs to be strengthened, because the concept of conditional convergence depends on the ordering of the index set.

If ${\displaystyle a:I\mapsto G}$ is a function from an index set ${\displaystyle I}$ to a set ${\displaystyle G,}$ then the "series" associated to ${\displaystyle a}$ is the formal sum of the elements ${\displaystyle a(x)\in G}$ over the index elements ${\displaystyle x\in I}$ denoted by the

$${\displaystyle \sum _{x\in I}a(x).}$$

When the index set is the natural numbers ${\displaystyle I=\mathbb {N} ,}$ the function ${\displaystyle a:\mathbb {N} \mapsto G}$ is a sequence denoted by ${\displaystyle a(n)=a_{n}.}$ A series indexed on the natural numbers is an ordered formal sum and so we rewrite ${\textstyle \sum _{n\in \mathbb {N} }}$ as ${\textstyle \sum _{n=0}^{\infty }}$ in order to emphasize the ordering induced by the natural numbers. Thus, we obtain the common notation for a series indexed by the natural numbers

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

When summing a family ${\displaystyle \left\{a_{i}:i\in I\right\}}$ of non-negative real numbers, define

$${\displaystyle \sum _{i\in I}a_{i}=\sup \left\{\sum _{i\in A}a_{i}\,:A\subseteq I,A{\text{ finite}}\right\}\in [0,+\infty ].}$$

When the supremum is finite then the set of ${\displaystyle i\in I}$ such that ${\displaystyle a_{i}>0}$ is countable. Indeed, for every ${\displaystyle n\geq 1,}$ the cardinality ${\displaystyle \left|A_{n}\right|}$ of the set ${\displaystyle A_{n}=\left\{i\in I:a_{i}>1/n\right\}}$ is finite because

$${\displaystyle {\frac {1}{n}}\,\left|A_{n}\right|=\sum _{i\in A_{n}}{\frac {1}{n}}\leq \sum _{i\in A_{n}}a_{i}\leq \sum _{i\in I}a_{i}<\infty .}$$

If ${\displaystyle I}$ is countably infinite and enumerated as ${\displaystyle I=\left\{i_{0},i_{1},\ldots \right\}}$ then the above defined sum satisfies

$${\displaystyle \sum _{i\in I}a_{i}=\sum _{k=0}^{+\infty }a_{i_{k}},}$$

provided the value ${\displaystyle \infty }$ 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.

Let ${\displaystyle a:I\to X}$ be a map, also denoted by ${\displaystyle \left(a_{i}\right)_{i\in I},}$ from some non-empty set ${\displaystyle I}$ into a Hausdorff abelian topological group ${\displaystyle X.}$ Let ${\displaystyle \operatorname {Finite} (I)}$ be the collection of all finite subsets of ${\displaystyle I,}$ with ${\displaystyle \operatorname {Finite} (I)}$ viewed as a directed set, ordered under inclusion ${\displaystyle \,\subseteq \,}$ with union as join. The family ${\displaystyle \left(a_{i}\right)_{i\in I},}$ is said to be unconditionally summable if the following limit, which is denoted by ${\displaystyle \sum _{i\in I}a_{i}}$ and is called the sum of ${\displaystyle \left(a_{i}\right)_{i\in I},}$ exists in ${\displaystyle X:}$

$${\displaystyle \sum _{i\in I}a_{i}:=\lim _{A\in \operatorname {Finite} (I)}\ \sum _{i\in A}a_{i}=\lim \left\{\sum _{i\in A}a_{i}\,:A\subseteq I,A{\text{ finite }}\right\}}$$

Saying that the sum ${\displaystyle S:=\sum _{i\in I}a_{i}}$ is the limit of finite partial sums means that for every neighborhood ${\displaystyle V}$ of the origin in ${\displaystyle X,}$ there exists a finite subset ${\displaystyle A_{0}}$ of ${\displaystyle I}$ such that

$${\displaystyle S-\sum _{i\in A}a_{i}\in V\qquad {\text{ for every finite superset}}\;A\supseteq A_{0}.}$$

Because ${\displaystyle \operatorname {Finite} (I)}$ is not totally ordered, this is not a limit of a sequence of partial sums, but rather of a net.[17][18]

For every neighborhood ${\displaystyle W}$ of the origin in ${\displaystyle X,}$ there is a smaller neighborhood ${\displaystyle V}$ such that ${\displaystyle V-V\subseteq W.}$ It follows that the finite partial sums of an unconditionally summable family ${\displaystyle \left(a_{i}\right)_{i\in I},}$ form a Cauchy net, that is, for every neighborhood ${\displaystyle W}$ of the origin in ${\displaystyle X,}$ there exists a finite subset ${\displaystyle A_{0}}$ of ${\displaystyle I}$ such that

$${\displaystyle \sum _{i\in A_{1}}a_{i}-\sum _{i\in A_{2}}a_{i}\in W\qquad {\text{ for all finite supersets }}\;A_{1},A_{2}\supseteq A_{0},}$$

which implies that ${\displaystyle a_{i}\in W}$ for every ${\displaystyle i\in I\setminus A_{0}}$ (by taking ${\displaystyle A_{1}:=A_{0}\cup \{i\}}$ and ${\displaystyle A_{2}:=A_{0}}$).

When ${\displaystyle X}$ is complete, a family ${\displaystyle \left(a_{i}\right)_{i\in I}}$ is unconditionally summable in ${\displaystyle X}$ if and only if the finite sums satisfy the latter Cauchy net condition. When ${\displaystyle X}$ is complete and ${\displaystyle \left(a_{i}\right)_{i\in I},}$ is unconditionally summable in ${\displaystyle X,}$ then for every subset ${\displaystyle J\subseteq I,}$ the corresponding subfamily ${\displaystyle \left(a_{j}\right)_{j\in J},}$ is also unconditionally summable in ${\displaystyle X.}$

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 ${\displaystyle X=\mathbb {R} .}$

If a family ${\displaystyle \left(a_{i}\right)_{i\in I}}$ in ${\displaystyle X}$ is unconditionally summable then for every neighborhood ${\displaystyle W}$ of the origin in ${\displaystyle X,}$ there is a finite subset ${\displaystyle A_{0}\subseteq I}$ such that ${\displaystyle a_{i}\in W}$ for every index ${\displaystyle i}$ not in ${\displaystyle A_{0}.}$ If ${\displaystyle X}$ is a first-countable space then it follows that the set of ${\displaystyle i\in I}$ such that ${\displaystyle a_{i}\neq 0}$ is countable. This need not be true in a general abelian topological group (see examples below).

Suppose that ${\displaystyle I=\mathbb {N} .}$ If a family ${\displaystyle a_{n},n\in \mathbb {N} ,}$ is unconditionally summable in a Hausdorff abelian topological group ${\displaystyle X,}$ then the series in the usual sense converges and has the same sum,

$${\displaystyle \sum _{n=0}^{\infty }a_{n}=\sum _{n\in \mathbb {N} }a_{n}.}$$

By nature, the definition of unconditional summability is insensitive to the order of the summation. When ${\displaystyle \sum a_{n}}$ is unconditionally summable, then the series remains convergent after any permutation ${\displaystyle \sigma$ :\mathbb {N} \to \mathbb {N} } of the set ${\displaystyle \mathbb {N} }$ of indices, with the same sum,

$${\displaystyle \sum _{n=0}^{\infty }a_{\sigma (n)}=\sum _{n=0}^{\infty }a_{n}.}$$

Conversely, if every permutation of a series ${\displaystyle \sum a_{n}}$ converges, then the series is unconditionally convergent. When ${\displaystyle X}$ is complete then unconditional convergence is also equivalent to the fact that all subseries are convergent; if ${\displaystyle X}$ is a Banach space, this is equivalent to say that for every sequence of signs ${\displaystyle \varepsilon _{n}=\pm 1}$, the series

$${\displaystyle \sum _{n=0}^{\infty }\varepsilon _{n}a_{n}}$$

converges in ${\displaystyle X.}$

If ${\displaystyle X}$ is a topological vector space (TVS) and ${\displaystyle \left(x_{i}\right)_{i\in I}}$ is a (possibly uncountable) family in ${\displaystyle X}$ then this family is summable[19] if the limit ${\displaystyle \lim _{A\in \operatorname {Finite} (I)}x_{A}}$ of the net ${\displaystyle \left(x_{A}\right)_{A\in \operatorname {Finite} (I)}}$ exists in ${\displaystyle X,}$ where ${\displaystyle \operatorname {Finite} (I)}$ is the directed set of all finite subsets of ${\displaystyle I}$ directed by inclusion ${\displaystyle \,\subseteq \,}$ and ${\textstyle x_{A}:=\sum _{i\in A}x_{i}.}$

It is called absolutely summable if in addition, for every continuous seminorm ${\displaystyle p}$ on ${\displaystyle X,}$ the family ${\displaystyle \left(p\left(x_{i}\right)\right)_{i\in I}}$ is summable. If ${\displaystyle X}$ is a normable space and if ${\displaystyle \left(x_{i}\right)_{i\in I}}$ is an absolutely summable family in ${\displaystyle X,}$ then necessarily all but a countable collection of ${\displaystyle x_{i}}$’s are zero. Hence, in normed spaces, it is usually only ever necessary to consider series with countably many terms.

Summable families play an important role in the theory of nuclear spaces.

The notion of series can be easily extended to the case of a seminormed space. If ${\displaystyle x_{n}}$ is a sequence of elements of a normed space ${\displaystyle X}$ and if ${\displaystyle x\in X}$ then the series ${\displaystyle \sum x_{n}}$ converges to ${\displaystyle x}$ in ${\displaystyle X}$ if the sequence of partial sums of the series ${\textstyle \left(\sum _{n=0}^{N}x_{n}\right)_{N=1}^{\infty }}$ converges to ${\displaystyle x}$ in ${\displaystyle X}$; to wit,

$${\displaystyle \left\|x-\sum _{n=0}^{N}x_{n}\right\|\to 0\quad {\text{ as }}N\to \infty .}$$

More generally, convergence of series can be defined in any abelian Hausdorff topological group. Specifically, in this case, ${\displaystyle \sum x_{n}}$ converges to ${\displaystyle x}$ if the sequence of partial sums converges to ${\displaystyle x.}$

If ${\displaystyle (X,|\cdot |)}$ is a seminormed space, then the notion of absolute convergence becomes: A series ${\textstyle \sum _{i\in I}x_{i}}$ of vectors in ${\displaystyle X}$ converges absolutely if

$${\displaystyle \sum _{i\in I}\left|x_{i}\right|<+\infty }$$

in which case all but at most countably many of the values ${\displaystyle \left|x_{i}\right|}$ are necessarily zero.

If a countable 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 Dvoretzky & Rogers (1950)).

Conditionally convergent series can be considered if ${\displaystyle I}$ is a well-ordered set, for example, an ordinal number ${\displaystyle \alpha _{0}.}$ In this case, define by transfinite recursion:

$${\displaystyle \sum _{\beta <\alpha +1}a_{\beta }=a_{\alpha }+\sum _{\beta <\alpha }a_{\beta }}$$

and for a limit ordinal ${\displaystyle \alpha ,}$

$${\displaystyle \sum _{\beta <\alpha }a_{\beta }=\lim _{\gamma \to \alpha }\sum _{\beta <\gamma }a_{\beta }}$$

if this limit exists. If all limits exist up to ${\displaystyle \alpha _{0},}$ then the series converges.

1. Given a function ${\displaystyle f:X\to Y}$ into an abelian topological group ${\displaystyle Y,}$ define for every ${\displaystyle a\in X,}$
   $${\displaystyle f_{a}(x)={\begin{cases}0&x\neq a,\\f(a)&x=a,\\\end{cases}}}$$

   

   a function whose support is a singleton ${\displaystyle \{a\}.}$ Then

   $${\displaystyle f=\sum _{a\in X}f_{a}}$$

   

   in the topology of pointwise convergence (that is, the sum is taken in the infinite product group ${\displaystyle Y^{X}}$).

2. In the definition of partitions of unity, one constructs sums of functions over arbitrary index set ${\displaystyle I,}$
   $${\displaystyle \sum _{i\in I}\varphi _{i}(x)=1.}$$

   

   While, formally, this requires a notion of sums of uncountable series, by construction there are, for every given ${\displaystyle x,}$ 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, that is, for every ${\displaystyle x}$ there is a neighborhood of ${\displaystyle x}$ in which all but a finite number of functions vanish. Any regularity property of the ${\displaystyle \varphi _{i},}$ 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 ${\displaystyle \omega _{1}}$ viewed as a topological space in the order topology, the constant function ${\displaystyle f:\left[0,\omega _{1}\right)\to \left[0,\omega _{1}\right]}$ given by ${\displaystyle f(\alpha )=1}$ satisfies
   $${\displaystyle \sum _{\alpha \in [0,\omega _{1})}f(\alpha )=\omega _{1}}$$

   

   (in other words, ${\displaystyle \omega _{1}}$ copies of 1 is ${\displaystyle \omega _{1}}$) only if one takes a limit over all countable partial sums, rather than finite partial sums. This space is not separable.