26 Facts About Infinite series

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.

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.

Generally, the terms of a Infinite series come from a ring, often the field of the real numbers or the field of the complex numbers.

An infinite series or simply a series is an infinite sum, represented by an infinite expression of the form.

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Such a series can be identified with a finite sum, so it is only infinite in a trivial sense.

Which is convergent, but the Infinite series formed by taking the absolute value of each term is the divergent harmonic Infinite series.

Similarly, if the ƒn are integrable on a closed and bounded interval I and converge uniformly, then the Infinite series is integrable on I and can be integrated term-by-term.

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.

Formal power Infinite 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–Poincare Infinite series is a formal power Infinite series used to study graded algebras.

In many cases, a Dirichlet Infinite series can be extended to an analytic function outside the domain of convergence by analytic continuation.

Infinite series 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 p.

Investigation of the validity of infinite series is considered to begin with Gauss in the 19th century.

Cauchy insisted on strict tests of convergence; he showed that if two Infinite series are convergent their product is not necessarily so, and with him begins the discovery of effective criteria.

Infinite series showed the necessity of considering the subject of continuity in questions of convergence.

Semi-convergent Infinite series were studied by Poisson, who gave a general form for the remainder of the Maclaurin formula.

Fourier series were being investigatedas the result of physical considerations at the same time thatGauss, Abel, and Cauchy were working out the theory of infiniteseries.

Euler had already given the formulas for determining the coefficients in the Infinite series;Fourier was the first to assert and attempt to prove the generaltheorem.

Dirichlet's treatment, of trigonometric Infinite series was the subject of criticism and improvement byRiemann, Heine, Lipschitz, Schlafli, anddu Bois-Reymond.

The difference is that an asymptotic Infinite series cannot be made to produce an answer as exact as desired, the way that convergent Infinite series can.

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.

When is unconditionally summable, then the Infinite series remains convergent after any permutation of the set of indices, with the same sum,.

When is complete then unconditional convergence is equivalent to the fact that all subInfinite series are convergent; if is a Banach space, this is equivalent to say that for every sequence of signs, the Infinite series.

Notion of Infinite series can be easily extended to the case of a seminormed space.

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Conditionally convergent Infinite series can be considered if is a well-ordered set, for example, an ordinal number In this case, define by transfinite recursion:.