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In mathematical analysis, the alternating series test proves that an alternating series is convergent when its terms decrease monotonically in absolute value and approach zero in the limit. The test was devised by Gottfried Leibniz and is sometimes known as Leibniz's test , Leibniz's rule , or the Leibniz criterion .
Like any series, an alternating series is a convergent series if and only if the sequence of partial sums of the series converges to a limit. The alternating series test guarantees that an alternating series is convergent if the terms a n converge to 0 monotonically, but this condition is not necessary for convergence.
The formula is a special case of the Euler–Boole summation formula for alternating series, providing yet another example of a convergence acceleration technique that can be applied to the Leibniz series. In 1992, Jonathan Borwein and Mark Limber used the first thousand Euler numbers to calculate π to 5,263 decimal places with the Leibniz ...
In mathematics, Dirichlet's test is a method of testing for the convergence of a series that is especially useful for proving conditional convergence. It is named after its author Peter Gustav Lejeune Dirichlet , and was published posthumously in the Journal de Mathématiques Pures et Appliquées in 1862.
In mathematics and numerical analysis, the van Wijngaarden transformation is a variant on the Euler transform used to accelerate the convergence of an alternating series. One algorithm to compute Euler's transform runs as follows:
Once such a sequence is found, a similar question can be asked with f(n) taking the role of 1/n, and so on. In this way it is possible to investigate the borderline between divergence and convergence of infinite series. Using the integral test for convergence, one can show (see below) that, for every natural number k, the series
Divergence can also be shown directly from the definition: an infinite series converges if and only if the sequence of partial sums converges to a limit, in which case that limit is the value of the infinite series.
A closely related topic is the approximation of functions by generalized Fourier series, that is, approximations based upon summation of a series of terms based upon orthogonal polynomials.