Series (mathematics)

Definition

For any sequence ( a n ) of rational numbers, real numbers, complex numbers, functions thereof, etc., the associated series is defined as the ordered formal sum

∑ n = 0 ∞ a n = a 0 + a 1 + a 2 + ⋯ .

The sequence of partial sums ( s k ) associated to a series ∑ n = 0 ∞ a n is defined for each k as the sum of the sequence ( a n ) from a 0 to a k

s k = ∑ n = 0 k a n = a 0 + a 1 + ⋯ + a k .

By definition the series ∑ n = 0 ∞ a n converges to a limit L if and only if the associated sequence of partial sums ( s k ) converges to L . This definition is usually written as

L = ∑ n = 0 ∞ a n ⇔ L = lim k → ∞ s k .

More generally, if a : I ↦ G is a function from an index set I to a set G, then the series associated to a is the formal sum of the elements a ( x ) ∈ G over the index elements x ∈ I denoted by the

∑ x ∈ I a ( x ) .

When the index set is the natural numbers I = N , the function a : N ↦ G is a sequence denoted by a ( n ) = a n . A series indexed on the natural numbers is an ordered formal sum and so we rewrite ∑ n ∈ N as ∑ n = 0 ∞ 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

∑ n = 0 ∞ a n = a 0 + a 1 + a 2 + ⋯ .

When the set G is a semigroup, the sequence of partial sums ( s k ) ⊂ G associated to a sequence ( a n ) ⊂ G is defined for each k as the sum of the terms a 0 , a 1 , ⋯ , a k

s k = ∑ n = 0 k a n = a 0 + a 1 + ⋯ + a k .

When the semigroup G is also a topological space, then the series ∑ n = 0 ∞ a n converges to an element L ∈ G if and only if the associated sequence of partial sums ( s k ) converges to L . This definition is usually written as

L = ∑ n = 0 ∞ a n ⇔ L = lim k → ∞ s k .

Convergent series

A series  ∑a_n  is said to 'converge' or to 'be convergent' when the sequence S_N of partial sums has a finite limit. If the limit of S_N is infinite or does not exist, the series is said to diverge. When the limit of partial sums exists, it is called the sum of the series

∑ n = 0 ∞ a n = lim N → ∞ S N = lim N → ∞ ∑ n = 0 N 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 non-zero is the essence of the study of series. Consider the example

1 + 1 2 + 1 4 + 1 8 + ⋯ + 1 2 n + ⋯ .

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, ½, ¼, 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 ½, we still have a piece of length ½ unmarked, so we can certainly mark the next ¼. 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

S / 2 = 1 + 1 2 + 1 4 + 1 8 + ⋯ 2 = 1 2 + 1 4 + 1 8 + 1 16 + ⋯ .

Therefore,

S − S / 2 = 1 ⇒ S = 2.

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

x = 0.111 …

we are talking, in fact, just about the series

∑ n = 1 ∞ 1 10 n .

But 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, it should offend no sensibilities if we make no distinction between 0.111… and ¹/₉. Less clear is the argument that 9 × 0.111… = 0.999… = 1, but it is not untenable when we consider that we can formalize the proof knowing only that limit laws preserve the arithmetic operations. See 0.999... for more.

Examples

A geometric series is one where each successive term is produced by multiplying the previous term by a constant number (called the common ratio in this context). Example:

In general, the geometric series converges if and only if | z | < 1 .

An Arithmetico-geometric sequence is a generalization of the geometric series, which has coefficients of the common ratio equal to the terms in an arithmetic series. Example:

The harmonic series is the series

The harmonic series is divergent.

An alternating series is a series where terms alternate signs. Examples:

and

The p-series

converges if r > 1 and diverges for r ≤ 1, which can be shown with the integral criterion described below in convergence tests. As a function of r, the sum of this series is Riemann's zeta function.

A telescoping series

converges if the sequence b_n converges to a limit L as n goes to infinity. The value of the series is then b₁ − L.

There are some elementary series whose convergence is not yet known/proven. For example, it is unknown whether the Flint Hills series

converges or not. The convergence depends on how well π can be approximated with rational numbers (which is unknown as of yet). More specifically, the values of n with large numerical contributions to the sum are the numerators of the continued fraction convergents of π , a sequence beginning with 1, 3, 22, 333, 355, 103993, ... (sequence A046947 in the OEIS). These are integers that are close to n π for some integer n, so that sin ⁡ n π is close to 0 and its reciprocal is large. Max A. Alekseyev proved in 2011 that if the series converges then the irrationality measure of π is lower than 2.5, which is much smaller than current known bound of 7.6063....

Calculus and partial summation as an operation on sequences

Partial summation takes as input a sequence, { a_n }, and gives as output another sequence, { S_N }. It is thus a unary operation on sequences. Further, this function is linear, and thus is a linear operator on the vector space of sequences, denoted Σ. The inverse operator is the finite difference operator, Δ. These behave as discrete analogs of integration and differentiation, only for series (functions of a natural number) instead of functions of a real variable. For example, the sequence {1, 1, 1, ...} has series {1, 2, 3, 4, ...} as its partial summation, which is analogous to the fact that ∫ 0 x 1 d t = x .

In computer science it is known as prefix sum.

Properties of series

Series are classified not only by whether they converge or diverge, but also by the properties of the terms a_n (absolute or conditional convergence); type of convergence of the series (pointwise, uniform); the class of the term a_n (whether it is a real number, arithmetic progression, trigonometric function); etc.

Non-negative terms

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 ∑a_n with non-negative terms converges if and only if the sequence S_N of partial sums is bounded.

For example, the series

∑ n = 1 ∞ 1 n 2

is convergent, because the inequality

1 n 2 ≤ 1 n − 1 − 1 n , n ≥ 2 ,

and a telescopic sum argument implies that the partial sums are bounded by 2.

Absolute convergence

A series

∑ n = 0 ∞ a n

is said to converge absolutely if the series of absolute values

∑ n = 0 ∞ | a n |

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.

Conditional convergence

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

∑ n = 1 ∞ ( − 1 ) n + 1 n = 1 − 1 2 + 1 3 − 1 4 + 1 5 − ⋯

which is convergent (and its sum is equal to 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 a_n are real and S is any real number, that one can find a reordering so that the reordered series converges with sum equal to S.

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

∑ a n = ∑ λ n b n

where the partial sums B_N = b₀ + ··· + b_n are bounded, λ_n has bounded variation, and lim λ_n B_n exists:

sup N | ∑ n = 0 N b n | < ∞ ,     ∑ | λ n + 1 − λ n | < ∞   and   λ n B n   converges,

then the series ∑ a_n is convergent. This applies to the pointwise convergence of many trigonometric series, as in

∑ n = 2 ∞ sin ⁡ ( n x ) ln ⁡ n

with 0 < x < 2π. Abel's method consists in writing 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 ∑ a_n to the absolutely convergent series

∑ ( λ n − λ n + 1 ) B n .

Convergence tests

n-th term test: If lim_n→∞ a_n ≠ 0 then the series diverges.

Comparison test 1 (see Direct comparison test): If ∑b_n  is an absolutely convergent series such that |a_n | ≤ C |b_n | for some number C  and for sufficiently large n , then ∑a_n  converges absolutely as well. If ∑|b_n | diverges, and |a_n | ≥ |b_n | for all sufficiently large n , then ∑a_n  also fails to converge absolutely (though it could still be conditionally convergent, e.g. if the a_n  alternate in sign).

Comparison test 2 (see Limit comparison test): If ∑b_n  is an absolutely convergent series such that |a_n+1 /a_n | ≤ |b_n+1 /b_n | for sufficiently large n , then ∑a_n  converges absolutely as well. If ∑|b_n | diverges, and |a_n+1 /a_n | ≥ |b_n+1 /b_n | for all sufficiently large n , then ∑a_n  also fails to converge absolutely (though it could still be conditionally convergent, e.g. if the a_n  alternate in sign).

Ratio test: If there exists a constant C < 1 such that |a_n+1/a_n|<C for all sufficiently large n, then ∑a_n 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_n|^1/n ≤ C for all sufficiently large n, then ∑a_n converges absolutely.

Integral test: if ƒ(x) is a positive monotone decreasing function defined on the interval [1, ∞) with ƒ(n) = a_n for all n, then ∑a_n converges if and only if the integral  ∫₁^∞ ƒ(x) dx is finite.

Cauchy's condensation test: If a_n is non-negative and non-increasing, then the two series  ∑a_n  and  ∑2^ka_(2^k) are of the same nature: both convergent, or both divergent.

Alternating series test: A series of the form ∑(−1)ⁿ a_n (with a_n ≥ 0) is called alternating. Such a series converges if the sequence a_n 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.

Series of functions

A series of real- or complex-valued functions

∑ n = 0 ∞ f n ( x )

converges pointwise on a set E, if the series converges for each x in E as an ordinary series of real or complex numbers. Equivalently, the partial sums

s N ( x ) = ∑ n = 0 N f n ( x )

converge to ƒ(x) as N → ∞ for each x ∈ E.

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 Nth partial sum,

| s N ( x ) − f ( x ) |  

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_n are 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, Abel's uniform convergence test, Dini's test, and the Cauchy criterion.

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 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

∫ E | s N ( x ) − f ( x ) | 2 d x → 0

as N → ∞.

A power series is a series of the form

∑ n = 0 ∞ 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

∑ n = 0 ∞ x n n !

is the Taylor series of 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. 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 sequences that are otherwise difficult to handle; this is the method of generating functions.

Laurent series

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

∑ n = − ∞ ∞ 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.

A Dirichlet series is one of the form

∑ n = 1 ∞ a n 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

ζ ( s ) = ∑ n = 1 ∞ 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 C ∖ { 1 }   with a simple pole at 1.

This series can be directly generalized to general Dirichlet series.

Trigonometric series

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

1 2 A 0 + ∑ n = 1 ∞ ( A n cos ⁡ n x + B n sin ⁡ n x ) .

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

Development of infinite series

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.

Mathematicians from Kerala, India studied infinite series around 1350 CE.

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.

Convergence criteria

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

1 + α β 1 ⋅ γ x + α ( α + 1 ) β ( β + 1 ) 1 ⋅ 2 ⋅ γ ( γ + 1 ) x 2 + ⋯

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

1 + m 1 ! x + m ( m − 1 ) 2 ! x 2 + ⋯

corrected certain of Cauchy's conclusions, and gave a completely scientific summation of the series for complex values of m and 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.

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 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.

Semi-convergence

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

F ( x ) = 1 n + 2 n + ⋯ + ( 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

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

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.

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 limits.

Series in Banach spaces

The notion of series can be easily extended to the case of a Banach space. If x_n is a sequence of elements of a Banach space X, then the series Σx_n converges to x ∈ X if the sequence of partial sums of the series tends to x; to wit,

∥ x − ∑ n = 0 N x n ∥ → 0

as N → ∞.

More generally, convergence of series can be defined in any abelian Hausdorff topological group. Specifically, in this case, Σx_n converges to x if the sequence of partial sums converges to x.

Summations over arbitrary index sets

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.

Families of non-negative numbers

When summing a family {a_i}, i ∈ I, of non-negative numbers, one may define

∑ i ∈ I a i = sup { ∑ i ∈ A a i | A  finite,  A ⊂ I } ∈ [ 0 , + ∞ ] .

When the supremum is finite, the set of i ∈ I such that a_i > 0 is countable. Indeed, for every n ≥ 1, the set A n = { i ∈ I : a i > 1 / n } is finite, because

1 n card ( A n ) ≤ ∑ i ∈ A n a i ≤ ∑ i ∈ I a i < ∞ .

If I  is countably infinite and enumerated as I = {i₀, i₁,...} then the above defined sum satisfies

∑ i ∈ I a i = ∑ k = 0 + ∞ a i k ,

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.

Abelian topological groups

Let a : I → X, where I  is any set and X  is an abelian Hausdorff topological group. Let F  be the collection of all finite subsets of I. Note that F  is a directed set ordered under inclusion with union as join. Define the sum S  of the family a as the limit

S = ∑ i ∈ I a i = lim { ∑ i ∈ A a i | A ∈ F }

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 A₀ of I  such that

S − ∑ i ∈ A a i ∈ V , A ⊃ A 0 .

Because F  is not totally ordered, this is not a limit of a sequence of partial sums, but rather of a net.

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_i, i ∈ I, form a Cauchy net, that is: for every W, neighborhood of 0 in X, there is a finite subset A₀ of I  such that

∑ i ∈ A 1 a i − ∑ i ∈ A 2 a i ∈ W , A 1 , A 2 ⊃ A 0 .

When X  is 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_i, i ∈ I, is unconditionally summable in X, then for every subset J ⊂ I, the corresponding subfamily a_j, 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 A₀ of I  such that a_i ∈ W  for every i not in A₀. If X  is first-countable, it follows that the set of i ∈ I  such that a_i ≠ 0 is countable. This need not be true in a general abelian topological group (see examples below).

Unconditionally convergent series

Suppose that I = N. If a family a_n, 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,

∑ n = 0 ∞ a n = ∑ n ∈ N a n .

By nature, the definition of unconditional summability is insensitive to the order of the summation. When ∑a_n is unconditionally summable, then the series remains convergent after any permutation σ of the set N of indices, with the same sum,

∑ n = 0 ∞ a σ ( n ) = ∑ n = 0 ∞ a n .

Conversely, if every permutation of a series ∑a_n 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_n = ±1, the series

∑ n = 0 ∞ ε n a n

converges in X. If X  is a Banach space, then one may define the notion of absolute convergence. A series ∑a_n of vectors in X  converges absolutely if

∑ n ∈ N ∥ a n ∥ < + ∞ .

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 Dvoretzky & Rogers (1950)).

Well-ordered sums

Conditionally convergent series can be considered if I is a well-ordered set, for example an ordinal number α₀. One may define by transfinite recursion:

∑ β < α + 1 a β = a α + ∑ β < α a β

and for a limit ordinal α,

∑ β < α a β = lim γ → α ∑ β < γ a β

if this limit exists. If all limits exist up to α₀, then the series converges.