On 19 Aug 2012, at 17:22, [email protected] wrote:
Citeren Bruno Marchal <[email protected]>:
On 18 Aug 2012, at 17:19, Roger wrote:
Hi Bruno Marchal
Yes, you can square the square root of any number to test its
accuracy,
but there are a variety of algorithms used to calculate pi.
Which is correct ? See
http://en.wikipedia.org/wiki/Pi
The value obtained is assumed to be true if the infinite series
used to calculate pi converges. But I would think that
many if not most infinite series should
converge. Which one is the right one ? Is there a unique solution ?
Most series would not converge. In this case they all converge to
Pi, as they have been designed for that. Some just converge more
quickly than others.
Bruno
http://iridia.ulb.ac.be/~marchal/
And even divergent series can be resummed to yield a finite answer,
sometimes even using just a few terms.
And there are many notions of convergence. Searching a job in England
Ramanujan, just to show his ability to compute, said that he could
compute the sum of all the natural numbers 1 + 2 + 3 + 4 + 5 + ...
which gives:
-1/12,
of course :) (*)
The crazy thing is that when you compute mass of a photon in string
theory, you are naturally lead to a sum of two terms, the first one
giving 1/12, and the second being 1 + 2 + 3 + 4 + ...
Bruno
(*) It is the value of the analytical continuation of the Rieman Zeta
function on -1. But it follows also naturally from convergence
criteria not involving the zeta function. Zeta(s) is the sum of all 1/
n^s, with n natural number ≠ 0, and it is equal to the product of all
1/(1-1/p^s)) with p primes, by a famous relation of Euler.
Saibal
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