Series (mathematics)
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures (such as in combinatorics) through generating functions. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance. For a long time, the idea that such a potentially infinite summation  could produce a finite result was considered paradoxical. This paradox was resolved using the concept of a limit during the 17th century. Zeno's paradox of Achilles and the tortoise illustrates this counterintuitive property of infinite sums: Achilles runs after a tortoise, but when he reaches the position of the tortoise at the beginning of the race, the tortoise has reached a second position; when he reaches this second position, the tortoise is at a third position, and so on. Zeno concluded that Achilles could "never" reach the tortoise, and thus that movement does not exist. Zeno divided the race  into infinitely many sub-races, each requiring a finite amount of time, so that the total time for Achilles to catch the tortoise is given by a series. The resolution of the paradox is that, although the series has an infinite number of terms, it has a finite sum, which gives the time necessary for Achilles to catch up with the tortoise. In modern terminology, any (ordered) infinite sequence formula_1 of terms (that is, numbers, functions, or anything that can be added) defines a series, which is the operation of adding the 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 formula_2 or, using the summation sign, formula_3 The infinite sequence of additions implied by a series cannot be effectively carried on (at least in a finite amount of time). However, if the set to which the terms and their finite sums belong has a notion of limit, it is sometimes possible to assign a value to a series, called the sum of the series. This value is the limit as tends to infinity (if  the limit exists) of the finite sums of the first terms of the series, which are called the th partial sums of the series. That is, formula_4 When this limit exists, one says that the series is convergent or summable, or that the sequence formula_1 is summable. In this case, the limit is called the sum of the series. Otherwise, the series is said to be divergent. The notation formula_6 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 formula_7 both the addition—the process of adding—and its result—the "sum" of and . Generally, the terms of a series come from a ring, often the field formula_8 of the real numbers or the field formula_9 of the complex numbers. In this case, the set of all series is itself a ring (and even an associative algebra), in which the addition consists of adding the series term by term, and the multiplication is the Cauchy product. Basic properties. An infinite series or simply a  series is an infinite sum, represented by an infinite expression of the form formula_10 where formula_11 is any ordered sequence of terms, such as numbers, functions, or anything else that can be added (an abelian group). This is an expression that is obtained from the list of terms formula_12 by laying them side by side, and conjoining them with the symbol "+". A series may also be represented by using summation notation, such as formula_13 If an abelian group 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 , 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 formula_14, its th partial sum is formula_15 By definition, the series formula_16 "converges" to the limit (or simply "sums" to ), if the sequence of its partial sums has a limit . In this case, one usually writes formula_17 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 is said to converge or to "be convergent" when the sequence of partial sums has a finite limit. 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 formula_18 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 formula_19 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 , it can be seen that formula_20 Therefore, formula_21 The idiom can be extended to other, equivalent notions of series. For instance, a recurring decimal, as in formula_22 encodes the series formula_23 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 , 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. π. formula_41 formula_42 Natural logarithm of 2. formula_43 formula_44 formula_45 formula_46 formula_47 formula_48 formula_49 Natural logarithm base "e". formula_50  formula_51 Calculus and partial summation as an operation on sequences. Partial summation takes as input a sequence, ("a"), and gives as output another sequence, ("S"). 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 formula_52 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 (absolute or conditional convergence); type of convergence of the series (pointwise, uniform); the class of the term a (whether it is a real number, arithmetic progression, trigonometric function); etc. Non-negative terms. When "a" is a non-negative real number for every "n", the sequence "S" of partial sums is non-decreasing. It follows that a series  Σ"a" with non-negative terms converges if and only if the sequence "S" of partial sums is bounded. For example, the series formula_53 is convergent, because the inequality formula_54 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. Absolute convergence. A series formula_55 "converges absolutely" if the series of absolute values formula_56 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 formula_57 which is convergent (and its sum is equal to formula_58), 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 formula_59 are real and formula_60 is any real number, that one can find a reordering so that the reordered series converges  with sum equal to formula_60. Abel's test is an important tool for handling semi-convergent series. If a series has the form formula_62 where the partial sums formula_63 are bounded, formula_64 has bounded variation, and formula_65 exists: formula_66 then the series formula_67 is convergent. This applies to the point-wise convergence of many trigonometric series, as in formula_68 with formula_69. Abel's method consists in writing formula_70, and in performing a transformation similar to integration by parts (called summation by parts), that relates the given series formula_67 to the absolutely convergent series formula_72 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 formula_73, there is an exact error evaluation. Set formula_74 to be the partial sum formula_75 of the given alternating series formula_60. Then the next inequality holds: formula_77 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. Matrix  exponential. For the matrix exponential: formula_78 the following error evaluation holds (scaling and squaring method): formula_79 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 formula_124 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 formula_125 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 "N"th partial sum, formula_126 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" 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 formula_127 as "N" → ∞. Power series. A power series is a series of the form formula_128 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 formula_129 is the Taylor series of formula_130 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". 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. Formal power series. 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. 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. 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 formula_131 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. Dirichlet series. A Dirichlet series is one of the form formula_132 where "s" is a complex number. For example, if all "a" are equal to 1, then the Dirichlet series is the Riemann zeta function formula_133 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 formula_134 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: formula_135 The most important example of a trigonometric series is the Fourier series of a function. History of the theory of infinite series. Development of infinite series. Greek 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 formula_136 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 formula_137 corrected certain of Cauchy's conclusions, and gave a completely scientific summation of the series for complex values of formula_138 and formula_139. 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 formula_140 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. Generalizations. 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. Summations over arbitrary index sets. Definitions may be given for sums over an arbitrary index set formula_141 There are two main differences with the usual notion of series: first, there is no specific order given on the set formula_142; second, this set formula_142 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 formula_144 is a function from an index set formula_142 to a set formula_146 then the "series" associated to formula_147 is the formal sum of the elements formula_148 over the index  elements formula_149 denoted by the formula_150 When the index set is the natural numbers formula_151 the function formula_152 is a sequence denoted by formula_153 A series indexed on the natural numbers is an ordered formal sum and so we rewrite formula_154 as formula_155 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 formula_156 Families of non-negative numbers. When summing a family formula_157 of non-negative real numbers, define formula_158 When the supremum is finite then the set of formula_159 such that formula_160 is  countable. Indeed, for every formula_161 the cardinality formula_162 of the set formula_163 is finite because formula_164 If formula_142 is countably infinite and enumerated as formula_166 then the above defined sum satisfies formula_167 provided the value formula_168 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 formula_169 be a map, also denoted by formula_170 from some non-empty set formula_142 into a Hausdorff abelian topological  group formula_172 Let formula_173 be the collection of all finite subsets of formula_174 with formula_173 viewed as a directed set, ordered under inclusion formula_176 with union as join. The family formula_170 is said to be if the following limit, which is denoted by formula_178 and is called the of formula_170 exists in formula_180 formula_181 Saying that the sum formula_182 is the limit of finite partial sums means that for every neighborhood formula_183 of the origin in formula_184 there exists a finite subset formula_185 of formula_142 such that formula_187 Because formula_173 is not totally ordered, this is not a limit of  a sequence of partial sums, but rather of a net. For every neighborhood formula_189 of the origin in formula_184 there is a smaller neighborhood formula_183 such that formula_192 It follows that the finite partial sums of an unconditionally summable family formula_170 form a , that is, for every neighborhood formula_189 of the origin in formula_184 there exists a finite subset formula_185 of formula_142 such that formula_198 which implies that formula_199 for every formula_200 (by taking formula_201 and formula_202). When formula_203 is complete, a family formula_204 is unconditionally summable in formula_203 if and only if the finite sums satisfy the latter Cauchy  net condition. When formula_203 is complete and formula_170 is unconditionally summable in formula_184 then for every subset formula_209 the corresponding subfamily formula_210 is also unconditionally summable in formula_172 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 formula_212 If a family formula_204 in formula_203 is unconditionally summable then for every neighborhood formula_189 of the origin in formula_184 there is a finite subset formula_217 such that formula_199 for every index formula_219 not in formula_220 If formula_203 is a first-countable space then  it follows that the set of formula_159 such that formula_223 is countable. This need not be true in a general abelian topological group (see examples below). Unconditionally convergent series. Suppose that formula_224 If a family formula_225 is unconditionally summable in a Hausdorff abelian topological group formula_184 then the series in the usual sense converges and has the same sum, formula_227 By nature, the definition of unconditional summability is insensitive to the order of the summation. When formula_228 is unconditionally summable, then the series remains convergent after any permutation formula_229 of the set formula_230 of indices, with the same sum, formula_231  Conversely, if every permutation of a series formula_228 converges, then the series is unconditionally convergent. When formula_203 is complete then unconditional convergence is also equivalent to the fact that all subseries are convergent; if formula_203 is a Banach space, this is equivalent to say that for every sequence of signs formula_235, the series formula_236 converges in formula_172 Series in topological vector spaces. If formula_203 is a topological vector space (TVS) and formula_239 is a (possibly uncountable) family in formula_203 then this family is summable if the limit formula_241 of the net formula_242 converges in formula_184 where formula_173 is the directed  set of all finite subsets of formula_142 directed by inclusion formula_176 and formula_247 It is called absolutely summable if in addition, for every continuous seminorm formula_248 on formula_184 the family formula_250 is summable. If formula_203 is a normable space and if formula_239 is an absolutely summable family in formula_184 then necessarily all but a countable collection of formula_254'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 formula_255 is a sequence of elements of a normed space formula_203 and if formula_257 then the series formula_258 converges to formula_139 in formula_203 if the sequence of partial sums of the series formula_261 converges to formula_139 in formula_203; to wit, formula_264 More generally, convergence of series can be defined in any abelian Hausdorff topological group. Specifically, in this case, formula_258 converges to formula_139 if the sequence of partial sums converges to formula_267 If formula_268 is a seminormed space, then the notion of absolute convergence becomes: A series formula_269  of vectors in formula_203 converges absolutely if formula_271 in which case all but at most countably many of the values formula_272 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 sums. Conditionally convergent series can be considered if formula_142 is a well-ordered set, for example, an ordinal number formula_274 In this case, define by transfinite recursion: formula_275 and for a limit ordinal formula_276 formula_277 if this limit exists. If all limits exist up to formula_278  then the series converges.