Basic propertiesAn infinite series or simply a series is an infinite sum, represented by an of the form where is any ordered of terms, such as s, , or anything else that can be (an ). This is an expression that is obtained from the list of terms by laying them side by side, and conjoining them with the symbol "+". A series may also be represented by using summation notation, such as If an abelian group of terms has a concept of (e.g., if it is a ), then some series, the , can be interpreted as having a value in , called the ''sum of the series''. This includes the common cases from , in which the group is the field of s or the field of s. Given a series , its th partial sum is By definition, the series ''converges'' to the limit (or simply ''sums'' to ), if the sequence of its partial sums has a limit . 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 seriesA series is said to or to ''be convergent'' when the sequence of partial sums has a finite . If the limit of is infinite or does not exist, the series is said to diverge. When the limit of partial sums exists, it is called the value (or sum) of the series An easy way that an infinite series can converge is if all the are zero for 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 It is possible to "visualize" its convergence on the : we can imagine a of length 2, with successive s 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 . If the series is denoted , it can be seen that Therefore, The idiom can be extended to other, equivalent notions of series. For instance, a , as in encodes the series Since these series always converge to (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 , which only relies on the fact that the limit laws for series preserve the ; for more detail on this argument, see 0.999....
Examples of numerical series* A '' '' is one where each successive term is produced by multiplying the previous term by a constant number (called the common ratio in this context). For example: In general, the geometric series converges , in which case it converges to
Natural logarithm of 2
Natural logarithm base ''e''
Calculus and partial summation as an operation on sequencesPartial summation takes as input a sequence, (''a''''n''), and gives as output another sequence, (''S''''N''). It is thus a on sequences. Further, this function is , and thus is a on the of sequences, denoted Σ. The inverse operator is the operator, denoted Δ. These behave as discrete analogues of and , 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 , it is known as .
Properties of seriesSeries are classified not only by whether they converge or diverge, but also by the properties of the terms an (absolute or conditional convergence); type of convergence of the series (pointwise, uniform); the class of the term an (whether it is a real number, arithmetic progression, trigonometric function); etc.
Non-negative termsWhen ''an'' is a non-negative real number for every ''n'', the sequence ''SN'' of partial sums is non-decreasing. It follows that a series Σ''an'' with non-negative terms converges if and only if the sequence ''SN'' of partial sums is bounded. For example, the series is convergent, because the inequality and a telescopic sum argument implies that the partial sums are bounded by 2. The exact value of the original series is the .
Absolute convergenceA series ''converges absolutely'' if the series of s 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 convergenceA 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 which is convergent (and its sum is equal to ), but the series formed by taking the absolute value of each term is the divergent . The 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 . is an important tool for handling semi-convergent series. If a series has the form 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 (called ), that relates the given series to the absolutely convergent series
Evaluation of truncation errorsThe evaluation of truncation errors is an important procedure in (especially and ).
When conditions of the alternating series test are satisfied by , there is an exact error evaluation. Set to be the partial sum of the given alternating series . Then the next inequality holds:
is a statement that includes the evaluation of the error term when the is truncated.
By using the , we can obtain the evaluation of the error term when the is truncated.
For the : the following error evaluation holds (scaling and squaring method):
Convergence testsThere exist many tests that can be used to determine whether particular series converge or diverge. * '' n-th term test'': If , then the series diverges; if , then the test is inconclusive. * Comparison test 1 (see ): If is an series such that for some number and for sufficiently large , then converges absolutely as well. If diverges, and for all sufficiently large , then also fails to converge absolutely (though it could still be conditionally convergent, for example, if the alternate in sign). * Comparison test 2 (see ): If is an absolutely convergent series such that for sufficiently large , then converges absolutely as well. If diverges, and for all sufficiently large , then also fails to converge absolutely (though it could still be conditionally convergent, for example, if the alternate in sign). * : If there exists a constant such that for all sufficiently large , then converges absolutely. When the ratio is less than , but not less than a constant less than , convergence is possible but this test does not establish it. * : If there exists a constant such that for all sufficiently large , then converges absolutely. * : if is a positive function defined on the interval _converges_if_and_only_if_the_integral.html" ;"title=",\infty) with for all , then converges if and only if the integral">,\infty) with for all , then converges if and only if the integral is finite. * Cauchy's condensation test: If is non-negative and non-increasing, then the two series and are of the same nature: both convergent, or both divergent. * Alternating series test: A series of the form (with ) is called ''alternating''. Such a series converges if the '''' is and converges to . The converse is in general not true. * For some specific types of series there are more specialized convergence tests, for instance for there is the .
Series of functionsA series of real- or complex-valued functions 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 converge to ''ƒ''(''x'') as ''N'' → ∞ for each ''x'' ∈ ''E''. A stronger notion of convergence of a series of functions is the . A series converges uniformly if it converges pointwise to the function ''ƒ''(''x''), and the error in approximating the limit by the ''N''th partial sum, 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 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 sequence, 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 null set, measure zero. Other modes of convergence depend on a different 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 as ''N'' → ∞.
Power series: A power series is a series of the form The 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 is the Taylor series of 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 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.
Formal power seriesWhile 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 to describe and study s that are otherwise difficult to handle, for example, using the method of s. 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 , 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 . In this case the algebra of formal power series is the total algebra of the monoid of natural numbers over the underlying term ring. 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 seriesLaurent 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 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.
Dirichlet series: A Dirichlet series is one of the form where ''s'' is a . For example, if all ''a''''n'' are equal to 1, then the Dirichlet series is the Riemann zeta function 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 with a simple pole (complex analysis), pole at 1. This series can be directly generalized to general Dirichlet series.
Trigonometric seriesA series of functions in which the terms are trigonometric functions is called a trigonometric series: The most important example of a trigonometric series is the of a function.
History of the theory of infinite series
Development of infinite seriesGreek 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. Mathematicians from Kerala, India studied infinite series around 1350 CE. 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 for all functions for which they exist was provided by Brook Taylor. Leonhard Euler in the 18th century, developed the theory of and q-series.
Convergence criteriaThe 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 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 corrected certain of Cauchy's conclusions, and gave a completely scientific summation of the series for complex values of and . 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 convergenceThe theory of 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-convergenceA series is said to be semi-convergent (or conditionally convergent) if it is convergent but not . 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 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.
Fourier serieswere 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 (''Journal für die reine und angewandte Mathematik, 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.
Asymptotic seriesAsymptotic 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 seriesUnder 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.
Summations over arbitrary index setsDefinitions may be given for sums over an arbitrary index set There are two main differences with the usual notion of series: first, there is no specific order given on the set ; second, this set 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 is a Function (mathematics), function from an index set to a set then the "series" associated to is the formal sum of the elements over the index elements denoted by the When the index set is the natural numbers the function is a denoted by A series indexed on the natural numbers is an ordered formal sum and so we rewrite as 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
Families of non-negative numbersWhen summing a family of non-negative real numbers, define When the supremum is finite then the set of such that is countable. Indeed, for every the cardinality of the set is finite because If is countably infinite and enumerated as then the above defined sum satisfies 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 groupsLet be a map, also denoted by from some non-empty set into is a Hausdorff space, Hausdorff Abelian group, abelian topological group Let be the collection of all Finite set, finite subsets of with viewed as a directed set, Partially ordered set, ordered under Inclusion (mathematics), inclusion with Union (set theory), union as Join (mathematics), join. The family is said to be if the following Limit of a net, limit, which is denoted by and is called the of exists in Saying that the sum is the limit of finite partial sums means that for every neighborhood of the origin in there exists a finite subset of such that Because is not Total order, totally ordered, this is not a limit of a sequence of partial sums, but rather of a Net (mathematics), net. For every neighborhood of the origin in there is a smaller neighborhood such that It follows that the finite partial sums of an unconditionally summable family form a , that is, for every neighborhood of the origin in there exists a finite subset of such that which implies that for every (by taking and ). When is Complete topological group, complete, a family is unconditionally summable in if and only if the finite sums satisfy the latter Cauchy net condition. When is complete and is unconditionally summable in then for every subset the corresponding subfamily is also unconditionally summable in 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 If a family in is unconditionally summable then for every neighborhood of the origin in there is a finite subset such that for every index not in If is a first-countable space then it follows that the set of such that is countable. This need not be true in a general abelian topological group (see examples below).
Unconditionally convergent seriesSuppose that If a family is unconditionally summable in a Hausdorff abelian topological group then the series in the usual sense converges and has the same sum, By nature, the definition of unconditional summability is insensitive to the order of the summation. When is unconditionally summable, then the series remains convergent after any permutation of the set of indices, with the same sum, Conversely, if every permutation of a series converges, then the series is unconditionally convergent. When is Complete topological group, complete then unconditional convergence is also equivalent to the fact that all subseries are convergent; if is a Banach space, this is equivalent to say that for every sequence of signs , the series converges in
Series in topological vector spacesIf is a topological vector space (TVS) and is a (possibly uncountable) family in then this family is summable if the limit of the Net (mathematics), net converges in where is the directed set of all finite subsets of directed by inclusion and It is called absolutely summable if in addition, for every continuous seminorm on the family is summable. If is a normable space and if is an absolutely summable family in then necessarily all but a countable collection of '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.
= Series in Banach and seminormed spaces= The notion of series can be easily extended to the case of a seminormed space. If is a sequence of elements of a normed space and if then the series converges to in if the sequence of partial sums of the series converges to in ; to wit, More generally, convergence of series can be defined in any Abelian group, abelian Hausdorff space, Hausdorff topological group. Specifically, in this case, converges to if the sequence of partial sums converges to If is a seminormed space, then the notion of absolute convergence becomes: A series of vectors in converges absolutely if in which case all but at most countably many of the values 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 ).
Well-ordered sumsConditionally convergent series can be considered if is a well-ordered set, for example, an ordinal number In this case, define by transfinite recursion: and for a limit ordinal if this limit exists. If all limits exist up to then the series converges.
Examples# Given a function into an abelian topological group define for every a function whose Support (mathematics), support is a Singleton (mathematics), singleton Then in the topology of pointwise convergence (that is, the sum is taken in the infinite product group ). # In the definition of partitions of unity, one constructs sums of functions over arbitrary index set While, formally, this requires a notion of sums of uncountable series, by construction there are, for every given 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 there is a neighborhood of in which all but a finite number of functions vanish. Any regularity property of the such as continuity, differentiability, that is preserved under finite sums will be preserved for the sum of any subcollection of this family of functions. # On the first uncountable ordinal viewed as a topological space in the order topology, the constant function given by satisfies (in other words, copies of 1 is ) only if one takes a limit over all ''countable'' partial sums, rather than finite partial sums. This space is not separable.
See also* Continued fraction * Convergence tests * Convergent series * Divergent series * Infinite compositions of analytic functions * Infinite expression (mathematics), Infinite expression * Infinite product * Iterated binary operation * List of mathematical series * Prefix sum * Sequence transformation * Series expansion
References*Thomas John I'Anson Bromwich, Bromwich, T. J. ''An Introduction to the Theory of Infinite Series'' MacMillan & Co. 1908, revised 1926, reprinted 1939, 1942, 1949, 1955, 1959, 1965. * * * * Walter Rudin, ''Principles of Mathematical Analysis'' (McGraw-Hill: New York, 1964). * * * * * *