Series (mathematics)

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In modern terminology, any (ordered) infinite sequence of terms (that is, numbers, functions, or anything that can be added) defines a series, which is the operation of adding the ai one after the other. To emphasize that there are an infinite number of terms, a series may be called an infinite series. Such a series is represented (or denoted) by an expression like

or, using the summation sign,



The notation denotes both the series—that is the implicit process of adding the terms one after the other indefinitely—and, if the series is convergent, the sum of the series—the result of the process. This is a generalization of the similar convention of denoting by both the addition—the process of adding—and its result—the sum of a and b.


Basic properties



If an abelian group A of terms has a concept of limit (e.g., if it is a metric space), then some series, the convergent series, can be interpreted as having a value in A, called the sum of the series. This includes the common cases from calculus, in which the group is the field of real numbers or the field of complex numbers. Given a series , its kth partial sum is [3]

By definition, the series converges to the limit L (or simply sums to L), if the sequence of its partial sums has a limit L. [4] In this case, one usually writes

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

Convergent series


A series an is said to converge or to be convergent when the sequence (sk) of partial sums has a finite limit. If the limit of sk is infinite or does not exist, the series is said to diverge. [5] [3] When the limit of partial sums exists, it is called the value (or sum) of the series


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



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

encodes the series

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....

Examples of numerical series


Natural logarithm of 2


Natural logarithm base e

Calculus and partial summation as an operation on sequences

Partial summation takes as input a sequence, (an), and gives as output another sequence, (SN). 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, denoted Δ. These behave as discrete analogues 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

In computer science, it is known as prefix sum.

Properties of series


Non-negative terms


For example, the series

is convergent, because the inequality


Absolute convergence

A series

converges absolutely if the series of absolute values


Conditional convergence


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


where the partial sums are bounded, has bounded variation, and exists:

then the series is convergent. This applies to the point-wise convergence of many trigonometric series, as in

with . Abel's method consists in writing , and in performing a transformation similar to integration by parts (called summation by parts), that relates the given series to the absolutely convergent series

Evaluation of truncation errors

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

Alternating series

When conditions of the alternating series test are satisfied by , there is an exact error evaluation. [10] Set to be the partial sum of the given alternating series . Then the next inequality holds:

Taylor series

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

Hypergeometric series

By using the ratio, we can obtain the evaluation of the error term when the hypergeometric series is truncated. [11]

Matrix exponential

For the matrix exponential:

the following error evaluation holds (scaling and squaring method): [12] [13] [14]

Convergence tests

There exist many tests that can be used to determine whether particular series converge or diverge.

Series of functions

A series of real- or complex-valued functions


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

A stronger notion of convergence of a series of functions is the uniform convergence. A series converges uniformly if it converges pointwise to the function ƒ(x), and the error in approximating the limit by the Nth partial sum,




as N  ∞.

Power series

A power series is a series of the form





Formal power series



Laurent series



Dirichlet series

A Dirichlet series is one of the form



This series can be directly generalized to general Dirichlet series.

Trigonometric series



History of the theory of infinite series

Development of infinite series


Mathematicians from Kerala, India studied infinite series around 1350 CE. [18]


Convergence criteria




Abel (1826) in his memoir on the binomial series




Uniform convergence





Genocchi (1852) has further contributed to the theory.


Fourier series




Asymptotic series


Divergent series



Summations over arbitrary index sets

Definitions may be given for sums over an arbitrary index set I. [19] 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. The notion of convergence needs to be strengthened, because the concept of conditional convergence depends on the ordering of the index set.



Families of non-negative numbers

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

When the supremum is finite, the set of i I such that ai > 0 is countable. Indeed, for every n ≥ 1, the set is finite, because

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



Abelian topological groups

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


Because F is not totally ordered, this is not a limit of a sequence of partial sums, but rather of a net. [20] [21]

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 ai, 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

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 ai, i I, is unconditionally summable in X, then for every subset J I, the corresponding subfamily aj, j J, is also unconditionally summable in X.


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 ai W for every i not in A0. If X is first-countable, it follows that the set of i I such that ai ≠ 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 an, 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,



converges in X.

Series in topological vector spaces

If X is a topological vector space (TVS) and is a (possibly uncountable) family in X then this family is summable [22] if the limit of the net converges in X, where is the directed set of all finite subsets of A directed by inclusion and .

It is called absolutely summable if in addition, for every continuous seminorm p on X, the family is summable. If X is a normable space and if is an absolutely summable family in X, then necessarily all but a countable collection of 's are 0. 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.

Series in Banach and semi-normed spaces

The notion of series can be easily extended to the case of a seminormed space. If xn is a sequence of elements of a normed space X and if x is in X, then the series ∑xn converges to x  in  X if the sequence of partial sums of the series converges to x in X; to wit,

as N  ∞.

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

If (X, |·|) is a semi-normed space, then the notion of absolute convergence becomes: A series of vectors in Xconverges absolutely if

in which case all but at most countably many of the values are necessarily zero.


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:

and for a limit ordinal α,



    a function whose support is a singleton {a}. Then

See also


  1. Thompson, Silvanus; Gardner, Martin (1998). Calculus Made Easy. ISBN   978-0-312-18548-0.
  2. "List of Calculus and Analysis Symbols". Math Vault. 2020-05-11. Retrieved 2020-08-30.
  3. 1 2 3 4 5 Weisstein, Eric W. "Series". Retrieved 2020-08-30.
  4. 1 2 Swokowski 1983 , p. 501
  5. Michael Spivak, Calculus
  6. "Infinite Series". Retrieved 2020-08-30.
  7. Gasper, G., Rahman, M. (2004). Basic hypergeometric series. Cambridge University Press.
  8. Max A. Alekseyev, On convergence of the Flint Hills series, arXiv:1104.5100, 2011.
  9. Weisstein, Eric W. "Flint Hills Series". MathWorld .
  10. Positive and Negative Terms: Alternating Series
  11. Johansson, F. (2016). Computing hypergeometric functions rigorously. arXiv preprint arXiv:1606.06977.
  12. Higham, N. J. (2008). Functions of matrices: theory and computation. Society for industrial and applied mathematics.
  13. Higham, N. J. (2009). The scaling and squaring method for the matrix exponential revisited. SIAM review, 51(4), 747-764.
  14. How and How Not to Compute the Exponential of a Matrix
  15. Nicolas Bourbaki (1989), Algebra, Springer: §III.2.11.
  18. "Indians predated Newton 'discovery' by 250 years".
  19. Jean Dieudonné, Foundations of mathematical analysis, Academic Press
  20. Bourbaki, Nicolas (1998). General Topology: Chapters 1–4. Springer. pp. 261–270. ISBN   978-3-540-64241-1.
  21. Choquet, Gustave (1966). Topology. Academic Press. pp. 216–231. ISBN   978-0-12-173450-3.
  22. Schaefer & Wolff 1999, pp. 179–180.

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MR 0033975