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A stronger notion of convergence of a series of functions is the uniform convergence. 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. 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.
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. 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.
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.
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.
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. 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.
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.
This series can be directly generalized to general Dirichlet series. A 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 Fourier series of a function. 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.
Mathematicians from Kerala, India studied infinite series around CE. In the 17th century, James Gregory worked in the new decimal system on infinite series and published several Maclaurin series.
In , 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. 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. It established simpler criteria of convergence, and the questions of remainders and the range of convergence.
Cauchy 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 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 in his memoir on the binomial series.
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 , who made the first elaborate investigation of the subject, of De Morgan from , whose logarithmic test DuBois-Reymond and Pringsheim have shown to fail within a certain region; of Bertrand , Bonnet , Malmsten , , the latter without integration ; Stokes , Paucker , Chebyshev , and Arndt General criteria began with Kummer , and have been studied by Eisenstein , Weierstrass in his various contributions to the theory of functions, Dini , DuBois-Reymond , and many others.
Pringsheim's memoirs present the most complete general theory. The theory of uniform convergence was treated by Cauchy , his limitations being pointed out by Abel, but the first to attack it successfully were Seidel and Stokes — Cauchy took up the problem again , acknowledging Abel's criticism, and reaching the same conclusions which Stokes had already found.
Thomae used the doctrine , 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.
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 , 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 , 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 Schlömilch Zeitschrift , Vol.
Genocchi has further contributed to the theory. 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 and his brother Johann Bernoulli and still earlier by Vieta.
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 —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 to attempt and for Dirichlet to handle in a thoroughly scientific manner see convergence of Fourier series.
Dirichlet's treatment Crelle , , of trigonometric series was the subject of criticism and improvement by Riemann , 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 , 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.
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