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 is the forward difference operator

and is a binomial coefficient.

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

where is the backward difference operator

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.