(aka Euler’s method, Euler transform, Euler summation)
Buchanan & Turner give a very simple derivation using difference operators and their basic properties.
Consider the summation of an alternating series:
Since is the RHS of , with , we have:
Substituting in :
Substituting in with gets the Euler transform:
Hamming in Numerical Methods gives a more complicated derivation using summation by parts, arriving at the more general form:
He says “the most frequent case of application is when ” (i.e. the form in equation ) and gives as an example of its use to make a series converge faster:
which converges more quickly.
(but e.g. with the series , in Hamming’s formula is , and the transformation converges no more quickly.)
Numerical Recipes says “Generally it is advisable to do a small number of terms directly, through term , and then apply the transformation to the rest of the series beginning with term . The formula (for even) is
the Euler transform is “the result of applying the binomial transform to the sequence associated with [a sequence’s] ordinary generating function.” – wikip, ‘binomial transform’
Buchanan, Turner – Numerical Methods and Analysis
Richard Hamming – Numerical methods for scientists and engineers
Numerical Recipes, 3rd ed., Ch.5