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The value of this limit, if it exists, is then the value of the series. If the limit of s k is infinite or does not exist, the series is said to diverge.

An easy way that an infinite series can converge is if all the a n are zero for n 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 non-zero is the essence of the study of series.

It is possible to "visualize" its convergence on the real number line: There is always room to mark the next segment, because the amount of line remaining is always the same as the last segment marked: 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 S , it can be seen that.

Mathematicians extend the idiom discussed earlier to other, equivalent notions of series. For instance, when we talk about a recurring decimal , as in.

But 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. It is thus a unary operation on sequences. These behave as discrete analogs of integration and differentiation , only for series functions of a natural number instead of functions of a real variable.

In computer science it is known as prefix sum. Series are classified not only by whether they converge or diverge, but also by the properties of the terms a n absolute or conditional convergence ; type of convergence of the series pointwise, uniform ; the class of the term a n whether it is a real number, arithmetic progression, trigonometric function ; etc.

When a n is a non-negative real number for every n , the sequence S N of partial sums is non-decreasing. The exact value of the original series is the Basel problem.

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.

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. Abel's test is an important tool for handling semi-convergent series. If a series has the form.

This applies to the pointwise convergence of many trigonometric series, as in. Equivalently, the partial sums. A stronger notion of convergence of a series of functions is called 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: 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.

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