An algorithm for easy implementation of Shanks’ non-linear transformation.

are the partial sums of a series .

Let , , and for , let

The are the equivalent of applying the th Shanks transformation to the sequence .

e.g. applied to the series

Result: Well, the best figure was – – 6 correct digits – and that used only the first 5 terms of the original series, at which point the original partial sum was – 1 correct digit only. That might be a fluke – but all the later terms have 4 correct digits, the worst is only 0.006% off. Maybe would be better to start the algorithm after a few terms, not right away.

p.s. 30,84,180,330 are of the form , see OEIS

**references**

Hamming – Numerical methods

P. Wynn – On a Device for Computing the Transformation, Math. Tables Aids Comp., vol. 10, pp. 91-96, 1956.