# the Euler transformation

Euler 1755.
(aka Euler’s method, Euler transform, Euler summation)

Derivation

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’

references

Buchanan, Turner – Numerical Methods and Analysis
Richard Hamming – Numerical methods for scientists and engineers
Numerical Recipes, 3rd ed., Ch.5