Euler's Transform

A technique for Series Convergence Improvement which takes a convergent alternating series

$$\sum_{k=0}^\infty (-1)^k a_k=a_0-a_1+a_2-\ldots$$ (1)

into a series with more rapid convergence to the same value to

$$s=\sum_{k=0}^\infty {(-1)^k\Delta^k a_0\over 2^{k+1}},$$ (2)

where the Forward Difference is defined by

$$\Delta^k a_0=\sum_{m=0}^k\equiv (-1)^m{k\choose m} a_{k-m}$$ (3)

(Abramowitz and Stegun 1972; Beeler et al. 1972, Item 120).

See also Forward Difference

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 16, 1972.

Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, Feb. 1972.