Re: A Formula for Pi

On Sat, 21 Jun 2008 21:37:06 -0700, Michael Press wrote:
[...] Mensanator <mensanator@xxxxxxx> wrote:
[...]
But I would prefer a series that converges in ~300 terms to the one
that converges in ~10**34 terms.

How about Ramanujan's series that gets eight decimal places for each
term?

1 2.sqrt{2} (4n)! [1103 + 26390n]
-- = --------- sum_n ------- ---------------
pi 9801 (n!)^4 (4*99)^4n

[Call this series R]

Also look up Euler's (who else?) transformation for accelerating
alternating series. In essence a difference table. Abramowitz and
Stegun, 3.6.27.

<http://mathworld.wolfram.com/EulersSeriesTransformation.html>

M. previously said he is using gmp rational arithmetic, so might
rule out R because of its sqrt(2) multiplier. On a different
tack, computing one term of R is as expensive as computing several
terms of Machin's formula. (Perhaps 13 multiplies, 1 divide, and
some adds per R term, vs. 2 multiplies, 1 divide, and some adds per
Machin term.)

mathworld doesn't seem to mention the Van Wijngaarden variant of Euler's
transform, http://en.wikipedia.org/wiki/Van_Wijngaarden_transformation

-jiw
.

Relevant Pages

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