Euler's Series Transformation

Euler's series transformation is a transformation that sometimes accelerates the rate of convergence for an alternating series. Given a convergent alternating series with sum


Abramowitz and Stegun (1972, p. 16) define Euler's transformation as


where Delta is the forward difference operator

 Delta^ka_0=sum_(m=0)^k(-1)^m(k; m)a_(k-m)

and (k; m) is a binomial coefficient.

An alternate formulation due to Knopp (1990, p. 244) instead defines the transformation as

 S=sum_(k=0)^infty(del ^ka_0)/(2^(k+1)),

where del is the backward difference operator

 del ^ka_0=sum_(m=0)^k(-1)^m(k; m)a_m.

Knopp (1990, p. 263) gives examples of different types of convergence behavior upon application of the transformation:


gives faster convergence,


gives same rate of convergence, and


gives slower convergence.

To see why the Euler transformation works, consider Knopp's convention for difference operator and write


Now repeat the process on the series in brackets to obtain


and continue to infinity. This proves each finite step in the derivation, although it doesn't actually prove the final step, since "continuing to infinity" involves use of a limit.

See also

Alternating Series, Series

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Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 16, 1972.Knopp, K. Theory and Application of Infinite Series. New York: Dover, 1990.

Referenced on Wolfram|Alpha

Euler's Series Transformation

Cite this as:

Weisstein, Eric W. "Euler's Series Transformation." From MathWorld--A Wolfram Web Resource.

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