 The Shanks transformation formula_26 already has two-digit accuracy, while the original partial sums only establish the same accuracy at formula_27 Remarkably, formula_28 has six digits accuracy, obtained from repeated Shank transformations applied to the first seven terms formula_29 As said before, formula_3 only obtains 6-digit accuracy after about summing 400,000 terms.The Shanks transformation is motivated by the observation that — for larger formula_25 — the partial sum formula_3 quite often behaves approximately as

with formula_34 so that the sequence converges transiently to the series result formula_7 for formula_8
So for formula_37 formula_25 and formula_39 the respective partial sums are