“Convergence Improvement”的概念、定义、习题解题方法及翻译-数学辞典

单词

Convergence Improvement

释义

Convergence Improvement

The improvement of the convergence properties of a Series, also called Convergence Acceleration, suchthat a Series reaches its limit to within some accuracy with fewer terms than required before.Convergence improvement can be effected by forming a linear combination with a Series whose sum isknown. Useful sums include

			(1)
			(2)
			(3)
			(4)

Kummer's transformation takes a convergent series

(5)

and another convergent series

(6)

with known

such that

(7)

Then a series with more rapid convergence to the same value is given by

(8)

(Abramowitz and Stegun 1972).

Euler's Transform takes a convergent alternating series

(9)

into a series with more rapid convergence to the same value to

(10)

where

(11)

(Abramowitz and Stegun 1972; Beeler et al. 1972, Item 120).

Given a series of the form

(12)

where

is an Analytic at 0 and on the closed unit Disk, and

(13)

then the series can be rearranged to


			(14)

where

(15)

is the Maclaurin Series of

and

is the Riemann Zeta Function (Flajolet and Vardi 1996). Thetransformed series exhibits geometric convergence. Similarly, if

is Analyticin

for some Positive Integer

, then

(16)

which converges geometrically (Flajolet and Vardi 1996). (16) can also be used to further accelerate the convergence ofseries (14).

See also Euler's Transform, Wilf-Zeilberger Pair

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 16, 1972.

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 288-289, 1985.

Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, Feb. 1972.

Flajolet, P. and Vardi, I. ``Zeta Function Expansions of Classical Constants.'' Unpublished manuscript. 1996. http://pauillac.inria.fr/algo/flajolet/Publications/landau.ps.

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