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Grandi's series

Index Grandi's series

In mathematics, the infinite series 1 - 1 + 1 - 1 + \dotsb, also written \sum_^ (-1)^n is sometimes called Grandi's series, after Italian mathematician, philosopher, and priest Guido Grandi, who gave a memorable treatment of the series in 1703. [1]

25 relations: Absolute convergence, Algebra, Cambridge University Press, Cesàro summation, Divergent geometric series, Divergent series, Eilenberg–Mazur swindle, History of calculus, Knot theory, Limit point, Luigi Guido Grandi, Mathematician, Mathematics, Permutation, Ramanujan summation, Series (mathematics), Telescoping series, Thomson's lamp, Undergraduate Texts in Mathematics, 1 + 1 + 1 + 1 + ⋯, 1 + 2 + 3 + 4 + ⋯, 1 + 2 + 4 + 8 + ⋯, 1 − 1 + 2 − 6 + 24 − 120 + ..., 1 − 2 + 3 − 4 + ⋯, 1 − 2 + 4 − 8 + ⋯.

Absolute convergence

In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite.

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Algebra

Algebra (from Arabic "al-jabr", literally meaning "reunion of broken parts") is one of the broad parts of mathematics, together with number theory, geometry and analysis.

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Cambridge University Press

Cambridge University Press (CUP) is the publishing business of the University of Cambridge.

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Cesàro summation

In mathematical analysis, Cesàro summation (also known as the Cesàro mean) assigns values to some infinite sums that are not convergent in the usual sense.

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Divergent geometric series

In mathematics, an infinite geometric series of the form is divergent if and only if | r | ≥ 1.

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Divergent series

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.

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Eilenberg–Mazur swindle

In mathematics, the Eilenberg–Mazur swindle, named after Samuel Eilenberg and Barry Mazur, is a method of proof that involves paradoxical properties of infinite sums.

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History of calculus

Calculus, known in its early history as infinitesimal calculus, is a mathematical discipline focused on limits, functions, derivatives, integrals, and infinite series.

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Knot theory

In topology, knot theory is the study of mathematical knots.

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Limit point

In mathematics, a limit point (or cluster point or accumulation point) of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contains a point of S other than x itself.

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Luigi Guido Grandi

Guido Grandi Dom Guido Grandi, O.S.B. Cam. (October 1, 1671 – July 4, 1742) was an Italian monk, priest, philosopher, theologian, mathematician, and engineer.

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Mathematician

A mathematician is someone who uses an extensive knowledge of mathematics in his or her work, typically to solve mathematical problems.

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Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

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Permutation

In mathematics, the notion of permutation relates to the act of arranging all the members of a set into some sequence or order, or if the set is already ordered, rearranging (reordering) its elements, a process called permuting.

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Ramanujan summation

Ramanujan summation is a technique invented by the mathematician Srinivasa Ramanujan for assigning a value to divergent infinite series.

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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.

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Telescoping series

In mathematics, a telescoping series is a series whose partial sums eventually only have a fixed number of terms after cancellation.

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Thomson's lamp

Thomson's lamp is a philosophical puzzle based on infinites.

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Undergraduate Texts in Mathematics

Undergraduate Texts in Mathematics (UTM) is a series of undergraduate-level textbooks in mathematics published by Springer-Verlag.

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1 + 1 + 1 + 1 + ⋯

In mathematics,, also written \sum_^ n^0, \sum_^ 1^n, or simply \sum_^ 1, is a divergent series, meaning that its sequence of partial sums does not converge to a limit in the real numbers.

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1 + 2 + 3 + 4 + ⋯

The infinite series whose terms are the natural numbers is a divergent series.

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1 + 2 + 4 + 8 + ⋯

In mathematics, is the infinite series whose terms are the successive powers of two.

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1 − 1 + 2 − 6 + 24 − 120 + ...

In mathematics, the divergent series was first considered by Euler, who applied summability methods to assign a finite value to the series.

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1 − 2 + 3 − 4 + ⋯

In mathematics, 1 − 2 + 3 − 4 + ··· is the infinite series whose terms are the successive positive integers, given alternating signs.

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1 − 2 + 4 − 8 + ⋯

In mathematics, is the infinite series whose terms are the successive powers of two with alternating signs.

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Redirects here:

1 − 1 + 1 − 1 + · · ·, 1-1+1-1+..., Grandi Series, Grandi series.

References

[1] https://en.wikipedia.org/wiki/Grandi's_series

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