|
Summability Methods
In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit. If a series converges, the individual terms of the series must approach zero. Thus any series in which the individual terms do not approach zero diverges. However, convergence is a stronger condition: not all series whose terms approach zero converge. A counterexample is the harmonic series :1 + \frac + \frac + \frac + \frac + \cdots =\sum_^\infty\frac. The divergence of the harmonic series was proven by the medieval mathematician Nicole Oresme. In specialized mathematical contexts, values can be objectively assigned to certain series whose sequences of partial sums diverge, in order to make meaning of the divergence of the series. A ''summability method'' or ''summation method'' is a partial function from the set of series to values. For example, Cesàro summation assigns Grandi's divergen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
 |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
Regularization (physics)
In physics, especially quantum field theory, regularization is a method of modifying observables which have singularities in order to make them finite by the introduction of a suitable parameter called the regulator. The regulator, also known as a "cutoff", models our lack of knowledge about physics at unobserved scales (e.g. scales of small size or large energy levels). It compensates for (and requires) the possibility of separation of scales that "new physics" may be discovered at those scales which the present theory is unable to model, while enabling the current theory to give accurate predictions as an "effective theory" within its intended scale of use. It is distinct from renormalization, another technique to control infinities without assuming new physics, by adjusting for self-interaction feedback. Regularization was for many decades controversial even amongst its inventors, as it combines physical and epistemological claims into the same equations. However, it i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
|
Convergent Series
In mathematics, a series is the sum of the terms of an infinite sequence of numbers. More precisely, an infinite sequence (a_1, a_2, a_3, \ldots) defines a series that is denoted :S=a_1 + a_2 + a_3 + \cdots=\sum_^\infty a_k. The th partial sum is the sum of the first terms of the sequence; that is, :S_n = a_1 +a_2 + \cdots + a_n = \sum_^n a_k. A series is convergent (or converges) if and only if the sequence (S_1, S_2, S_3, \dots) of its partial sums tends to a limit; that means that, when adding one a_k after the other ''in the order given by the indices'', one gets partial sums that become closer and closer to a given number. More precisely, a series converges, if and only if there exists a number \ell such that for every arbitrarily small positive number \varepsilon, there is a (sufficiently large) integer N such that for all n \ge N, :\left , S_n - \ell \right , 1 produce a convergent series: *: ++++++\cdots = . * Alternating the signs of reciprocals of powers o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
|
Properties Of Summation Methods
Property is the ownership of land, resources, improvements or other tangible objects, or intellectual property. Property may also refer to: Philosophy and science * Property (philosophy), in philosophy and logic, an abstraction characterizing an object * Material properties, properties by which the benefits of one material versus another can be assessed * Chemical property, a material's properties that becomes evident during a chemical reaction *Physical property, any property that is measurable whose value describes a state of a physical system *Thermodynamic properties, in thermodynamics and materials science, intensive and extensive physical properties of substances * Mathematical property, a property is any characteristic that applies to a given set * Semantic property *Mental property, a property of the mind studied by many sciences and parasciences Computer science * Property (programming), a type of class member in object-oriented programming * .properties, a Java Propert ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
 |
1 + 2 + 3 + 4 + ⋯
The infinite series whose terms are the positive integers is a divergent series. The ''n''th partial sum of the series is the triangular number \sum_^n k = \frac, which increases without bound as ''n'' goes to infinity. Because the sequence of partial sums fails to Limit of a sequence, converge to a finite limit, the series (mathematics), series does not have a sum. Although the series seems at first sight not to have any meaningful value at all, it can be manipulated to yield a number of different mathematical results. For example, many summation methods are used in mathematics to assign numerical values even to a divergent series. In particular, the methods of zeta function regularization and Ramanujan summation assign the series a value of , which is expressed by a famous formula: 1 + 2 + 3 + 4 + \cdots = -\frac, where the left-hand side has to be interpreted as being the value obtained by using one of the aforementioned summation methods and not as the sum of an infinite ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
 |
1 + 1 + 1 + 1 + ⋯
In mathematics, , also written , , or simply , is a divergent series. Nevertheless, it is sometimes imputed to have a value of , especially in physics. This value can be justified by certain mathematical methods for obtaining values from divergent series, including zeta function regularization. As a divergent series is a divergent series, meaning that its sequence of partial sums does not converge to a limit in the real numbers. The sequence 1 can be thought of as a geometric series with the common ratio 1. For some other divergent geometric series, including Grandi's series with ratio −1, and the series 1 + 2 + 4 + 8 + ⋯ with ratio 2, one can use the general solution for the sum of a geometric series with base 1 and ratio , obtaining , but this summation method fails for , producing a division by zero. Together with Grandi's series, this is one of two geometric series with rational ratio that diverges both for the real numbers and for all systems of -adic n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
 |
1 + 2 + 4 + 8 + ⋯
In mathematics, is the infinite series whose terms are the successive powers of two. As a geometric series, it is characterized by its first term, 1, and its common ratio, 2. As a series of real numbers it diverges to infinity, so in the usual sense it has no sum. However, it can be manipulated to yield a number of mathematically interesting results. For example, many summation methods are used in mathematics to assign numerical values even to divergent series. In particular, the Ramanujan summation of this series is −1, which is the limit of the series using the 2-adic metric. Summation The partial sum of the first n terms of 1 + 2 + 4 + 8 + \cdots is \sum_^ 2^k = 2^0+2^1 + \cdots + 2^ = 2^n - 1. Since the sequence 1, 3, 7, 15, \ldots of these partial sums diverges to infinity, so does the series. Therefore, any totally regular summation method gives a sum of infinity, including the Cesàro sum and Abel sum. On the other hand, there is at least one generally useful ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
 |
1 − 2 + 4 − 8 + ⋯
In mathematics, is the infinite series whose terms are the successive powers of two with alternating signs. As a geometric series, it is characterized by its first term, 1, and its common ratio, −2. : \sum_^ (-2)^k As a series of real numbers, it diverges. So in the usual sense it has no sum. In p-adic analysis, the series is associated with another value besides ∞, namely , which is the limit of the series using the 2-adic metric. Historical arguments Gottfried Leibniz considered the divergent alternating series as early as 1673. He argued that by subtracting either on the left or on the right, one could produce either positive or negative infinity, and therefore both answers are wrong and the whole should be finite: Now normally nature chooses the middle if neither of the two is permitted, or rather if it cannot be determined which of the two is permitted, and the whole is equal to a finite quantity Leibniz did not quite assert that the series had a ''sum'', but he di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
 |
1 − 1 + 2 − 6 + 24 − 120 + ⋯
In mathematics, :\sum_^\infty (-1)^k k! is a divergent series, first considered by Euler, that sums the factorials of the natural numbers with alternating signs. Despite being divergent, it can be assigned a value of approximately 0.596347 by Borel summation. Euler and Borel summation This series was first considered by Euler, who applied summability methods to assign a finite value to the series. The series is a sum of factorials that are alternately added or subtracted. One way to assign a value to this divergent series is by using Borel summation, where one formally writes :\sum_^\infty (-1)^k k! = \sum_^\infty (-1)^k \int_0^\infty x^k e^ \, dx. If summation and integration are interchanged (ignoring that neither side converges), one obtains: :\sum_^\infty (-1)^k k! = \int_0^\infty \left sum_^\infty (-x)^k \right^ \, dx. The summation in the square brackets converges when , x, < 1, and for those values equals . The |
 |
1 − 2 + 3 − 4 + ⋯
In mathematics, 1 − 2 + 3 − 4 + ··· is an Series (mathematics), infinite series whose terms are the successive positive integers, given alternating series, alternating signs. Using summation, sigma summation notation the sum of the first ''m'' terms of the series can be expressed as \sum_^m n(-1)^. The infinite series divergent series, diverges, meaning that its sequence of partial sums, , does not tend towards any finite Limit of a sequence, limit. Nonetheless, in the mid-18th century, Leonhard Euler wrote what he admitted to be a List of paradoxes#Mathematics, paradoxical equation: 1-2+3-4+\cdots=\frac. A Rigour#Mathematical rigour, rigorous explanation of this equation would not arrive until much later. Starting in 1890, Ernesto Cesàro, Émile Borel and others investigated well-defined methods to assign generalized sums to divergent series—including new interpretations of Euler's attempts. Many of these summability methods easily assign to a "value" of . Cesàro su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
Ferdinand Georg Frobenius
Ferdinand Georg Frobenius (26 October 1849 – 3 August 1917) was a German mathematician, best known for his contributions to the theory of elliptic functions, differential equations, number theory, and to group theory. He is known for the famous determinantal identities, known as Frobenius–Stickelberger formulae, governing elliptic functions, and for developing the theory of biquadratic forms. He was also the first to introduce the notion of rational approximations of functions (nowadays known as Padé approximants), and gave the first full proof for the Cayley–Hamilton theorem. He also lent his name to certain differential-geometric objects in modern mathematical physics, known as Frobenius manifolds. Biography Ferdinand Georg Frobenius was born on 26 October 1849 in Charlottenburg, a suburb of Berlin, from parents Christian Ferdinand Frobenius, a Protestant parson, and Christine Elizabeth Friedrich. He entered the Joachimsthal Gymnasium in 1860 when he was nearly el ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
|
Ernesto Cesàro
Ernesto Cesàro (12 March 1859 – 12 September 1906) was an Italian mathematician who worked in the field of differential geometry. He wrote a book, ''Lezioni di geometria intrinseca'' (Naples, 1890), on this topic, in which he also describes fractal, space-filling curves, partly covered by the larger class of de Rham curves, but are still known today in his honor as Cesàro curves. He is known also for his 'averaging' method for the 'Cesàro-summation' of divergent series, known as the Cesàro mean. Biography After a rather disappointing start of his academic career and a journey through Europe—with the most important stop at Liège, where his older brother Giuseppe Raimondo Pio Cesàro was teaching mineralogy at the local university—Ernesto Cesàro graduated from the University of Rome in 1887, while he was already part of the Royal Science Society of Belgium for the numerous works that he had already published. The following year, he obtained a mathematics chair at t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |