Shift rule

Last updated May 21, 2022

The shift rule is a mathematical rule for sequences and series.

For sequences, the rule states that if $(a_{n})$ is a sequence, then it converges if and only if $(a_{n+N})$ also converges, and in this case both sequences always converge to the same number.^[1]

For series, the rule states that the series $\sum \limits _{n=1}^{\infty }a_{n}$ converges to a number if and only if $\sum \limits _{n=1}^{\infty }a_{n+N}$ converges.^[2]

Related Research Articles

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. Series are used in most areas of mathematics, even for studying finite structures through generating functions. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance.

In mathematics, real analysis is the branch of mathematical analysis that studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrability.

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members. The number of elements is called the length of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function from natural numbers to the elements at each position. The notion of a sequence can be generalized to an indexed family, defined as a function from an index set that may not be numbers to another set of elements.

In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions $converges uniformly to a limiting function on a set if, given any arbitrarily small positive number, a number can be found such that each of the functions differ from by no more than at every point in . Described in an informal way, if converges to uniformly, then the rate at which approaches is "uniform" throughout its domain in the following sense: in order to guarantee that falls within a certain distance of, we do not need to know the value of in question — there can be found a single value of independent of, such that choosing will ensure that is within of for all . In contrast, pointwise convergence of to merely guarantees that for any given in advance, we can find so that, for that particular, falls within of whenever .$

In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or $. When it is positive, the power series converges absolutely and uniformly on compact sets inside the open disk of radius equal to the radius of convergence, and it is the Taylor series of the analytic function to which it converges. In case of multiple singularities of a function, the radius of convergence is the shortest or minimum of all the respective distances calculated from the center of the disk of convergence to the respective singularities of the function.$

In mathematics, the limit of a sequence is the value that the terms of a sequence "tend to", and is often denoted using the $symbol. If such a limit exists, the sequence is called convergent . A sequence that does not converge is said to be divergent . The limit of a sequence is said to be the fundamental notion on which the whole of mathematical analysis ultimately rests.$

In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which almost everywhere convergence of a sequence of functions implies convergence in the L¹ norm. Its power and utility are two of the primary theoretical advantages of Lebesgue integration over Riemann integration.

In mathematics, the ratio test is a test for the convergence of a series

In mathematics, the Weierstrass M-test is a test for determining whether an infinite series of functions converges uniformly and absolutely. It applies to series whose terms are bounded functions with real or complex values, and is analogous to the comparison test for determining the convergence of series of real or complex numbers. It is named after the German mathematician Karl Weierstrass (1815-1897).

In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to a given function as the argument of the function tends towards a particular, often infinite, point. Investigations by Dingle (1973) revealed that the divergent part of an asymptotic expansion is latently meaningful, i.e. contains information about the exact value of the expanded function.

In mathematics, more specifically in mathematical analysis, the Cauchy product is the discrete convolution of two infinite series. It is named after the French mathematician Augustin-Louis Cauchy.

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.

In mathematics, the root test is a criterion for the convergence of an infinite series. It depends on the quantity

In mathematical analysis, the alternating series test is the method used to show that an alternating series is convergent when its terms (1) decrease in absolute value, and (2) approach zero in the limit. The test was used by Gottfried Leibniz and is sometimes known as Leibniz's test, Leibniz's rule, or the Leibniz criterion. The test is only sufficient, not necessary, so some convergent alternating series may fail the first part of the test.

In mathematics, a series is the sum of the terms of an infinite sequence of numbers. More precisely, an infinite sequence $defines a series S that is denoted$

In mathematics, the Riesz–Fischer theorem in real analysis is any of a number of closely related results concerning the properties of the space L² of square integrable functions. The theorem was proven independently in 1907 by Frigyes Riesz and Ernst Sigismund Fischer.

The Cauchy convergence test is a method used to test infinite series for convergence. It relies on bounding sums of terms in the series. This convergence criterion is named after Augustin-Louis Cauchy who published it in his textbook Cours d'Analyse 1821.

In mathematics, convergence tests are methods of testing for the convergence, conditional convergence, absolute convergence, interval of convergence or divergence of an infinite series $.$

In mathematics, the nth-term test for divergence is a simple test for the divergence of an infinite series:

If $or if the limit does not exist, then diverges.$

In mathematics, for a sequence of complex numbers a₁, a₂, a₃, ... the infinite product

References

↑ Ueltschi, Daniel (2011), Analysis –MA131 (PDF), University of Warwick, p. 31.
↑ Alcock, Lara (2014), How to Think About Analysis, Oxford University Press, p. 102, ISBN 9780191035371 .

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[1] Ueltschi, Daniel (2011), Analysis –MA131 (PDF), University of Warwick, p. 31.

[2] Alcock, Lara (2014), How to Think About Analysis, Oxford University Press, p. 102, ISBN 9780191035371 .

[1]

[2]