[seqfan] Re: Another BBP formula for ln(2) ?
Simon Plouffe
simon.plouffe at gmail.com
Mon Jul 6 00:55:51 CEST 2009
Hello,
yes, it is true, by changing some variable, there is
several ways to get a more efficient formula than
log(2) = sum(1/n/2^n,n=1..infinity), (formula 1)
a question : what is C in equation 0.135 ? it must be Pi
isn't ?
For Alexander, look at equation 0.129 and 0.130 of
mr Mathar paper, these formulas are with the index 16^n :
it means that , formulas with an index of 2^n and 10 or
more terms are of very little interest since it converges
not faster than the standard log(2) series of Euler.
If I remember well, the formula (1) can be obtained
by using EULER transform of the series 1 - 1/2 + 1/3 - 1/4 ...= log(2).
Since there are many series of that later type : many others
can be obtained.
Also, and finally, there is always a way to SPLIT into 2 parts
a series of harmonic type to obtain a formula S = S1 + S2 and
deduce a NEW formula more rapidly convergent and for log(2)
there is a plethora of ways to get a better index than of 2^n.
best regards,
Simon Plouffe
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