# Re: Is convergence a unique test for pi ?

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On 19 Aug 2012, at 17:22, smi...@zonnet.nl wrote:```
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```Citeren Bruno Marchal <marc...@ulb.ac.be>:

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On 18 Aug 2012, at 17:19, Roger wrote:

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```Hi Bruno Marchal

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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 ?
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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.
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Bruno

http://iridia.ulb.ac.be/~marchal/

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And even divergent series can be resummed to yield a finite answer, sometimes even using just a few terms.
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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:
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-1/12,

of course :)   (*)

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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 + ...
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Bruno

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(*) 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.
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Saibal

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http://iridia.ulb.ac.be/~marchal/

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