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
(1)
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Abramowitz and Stegun (1972, p. 16) define Euler's transformation as
(2)
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where is the forward difference operator
(3)
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and is a binomial coefficient.
An alternate formulation due to Knopp (1990, p. 244) instead defines the transformation as
(4)
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where is the backward difference operator
(5)
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Knopp (1990, p. 263) gives examples of different types of convergence behavior upon application of the transformation:
(6)
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gives faster convergence,
(7)
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gives same rate of convergence, and
(8)
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gives slower convergence.
To see why the Euler transformation works, consider Knopp's convention for difference operator and write
(9)
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(10)
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Now repeat the process on the series in brackets to obtain
(11)
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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.