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

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.