Euler transform

Summary

The Euler transform rewrites an alternating series $\sum (-1)^n a_n$ as a new series built from the iterated differences of the $a_n$, with weights $2^{-n-1}$. When the $a_n$ vary smoothly, the new series converges much faster, often geometrically. It is a classical tool of series acceleration, and the same formula gives the Euler summation method for some divergent series.