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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

 S=sum_(k=0)^infty(-1)^ka_k,
(1)

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

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

where Delta is the forward difference operator

 Delta^ka_0=sum_(m=0)^k(-1)^m(k; m)a_(k-m)
(3)

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)),
(4)

where del is the backward difference operator

 del ^ka_0=sum_(m=0)^k(-1)^m(k; m)a_m.
(5)

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

 sum_(n=0)^infty((-1)^n)/(2^n)=1/2sum_(n=0)^infty1/(4^n)
(6)

gives faster convergence,

 sum_(n=0)^infty((-1)^n)/(3^n)=1/2sum_(n=0)^infty1/(3^n)
(7)

gives same rate of convergence, and

 sum_(n=0)^infty((-1)^n)/(4^n)=1/2sum_(n=0)^infty(3/8)^n
(8)

gives slower convergence.

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

S=u_0-u_1+u_2-...
(9)
=1/2u_0+1/2[(u_0-u_1)-(u_1-u_2)+(u_2-u_3)-...].
(10)

Now repeat the process on the series in brackets to obtain

 S=1/2u_0+1/4(u_0-u_1)+1/4[(u_0-2u_1+u_2)-(u_1-2u_2+u_3)+(u_2-2u_3+u_4)-...],
(11)

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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References

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. https://mathworld.wolfram.com/EulersSeriesTransformation.html

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