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)

Abramowitz and Stegun (1972, p. 16) define Euler's transformation as
(2)

where is the forward difference operator
(3)

and is a binomial coefficient.
An alternate formulation due to Knopp (1990, p. 244) instead defines the transformation as
(4)

where is the backward difference operator
(5)

Knopp (1990, p. 263) gives examples of different types of convergence behavior upon application of the transformation:
(6)

gives faster convergence,
(7)

gives same rate of convergence, and
(8)

gives slower convergence.
To see why the Euler transformation works, consider Knopp's convention for difference operator and write
(9)
 
(10)

Now repeat the process on the series in brackets to obtain
(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.