Euler's Series Transformation

Accelerates the rate of Convergence for an Alternating Series

$$S=\sum_{s=0}^\infty (-1)^s u_s = u_0-u_1+u_2-\ldots-u_{n-1}+\sum_{s=0}^\infty {(-1)^2\over 2^{s+1}} [\Delta^s u_n]$$ (1)

for $n$ Even and $\Delta$ the Forward Difference operator

$$\Delta^k u_n \equiv \sum_{m=0}^k (-1)^m {k\choose m} u_{n+k-m},$$ (2)

where ${k\choose m}$ are Binomial Coefficients. The Positive terms in the series can be converted to an Alternating Series using

$$\sum_{r=1}^\infty v_r = \sum_{r=1}^\infty (-1)^{r-1} w_r,$$ (3)

where

$$w_r\equiv v_r+2v_{2r}+4v_{4r}+8v_{8r}+\ldots.$$ (4)

See also Alternating Series

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.