The improvement of the convergence properties of a Series, also called Convergence Acceleration, such
that 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 is
known. Useful sums include
(1) | |||
(2) | |||
(3) | |||
(4) |
(5) |
(6) |
(7) |
(8) |
Euler's Transform takes a convergent alternating series
(9) |
(10) |
(11) |
Given a series of the form
(12) |
(13) |
(14) |
(15) |
(16) |
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 Unpublished manuscript. 1996.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.
© 1996-9 Eric W. Weisstein