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
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(1)
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Abramowitz and Stegun (1972, p. 16) define Euler's transformation as
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(2)
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where 
is the forward difference operator
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(3)
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and 
is a binomial coefficient.
An alternate formulation due to Knopp (1990, p. 244) instead defines the transformation as
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(4)
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where 
is the backward difference operator
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(5)
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Knopp (1990, p. 263) gives examples of different types of convergence behavior upon application of the transformation:
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(6)
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gives faster convergence,
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(7)
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gives same rate of convergence, and
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(8)
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gives slower convergence.
To see why the Euler transformation works, consider Knopp's convention for difference operator and write
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(9)
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 |  | ![1/2u_0+1/2[(u_0-u_1)-(u_1-u_2)+(u_2-u_3)-...].](/images/equations/EulersSeriesTransformation/Inline9.svg) |
(10)
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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)-...],](/images/equations/EulersSeriesTransformation/NumberedEquation9.svg) |
(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.
