Calculation of the decimals of Pi: Some new with old stuff!
January 4, 2009
Archimedes' method for the calculation of the decimals of . This approach mixes algorithms and series and allows in theory to obtain a convergence speed of as fast as we want it. The convergence stays for the moment linear.
A modern version of Achimedes algorithms consiste of defining the algorithm and
which allows to calculate given that
The efficience of this method is not bad since the convergence is in . It is not extraordinary either and numerous other methods (algorithms, series...) also converges linearly, equlas it or are an improvement (without taking into account the algorithms by Salamin-Brent or the other Borwein's brother of course). We offer still to start from this old idea and to calculate the decimals of with a speed of how great we choose it to be (according to a pre calculation). Hence, for all values of we have the following formula which stay valid:
We ask you to exactly calculate the strippy area below, which is a sector of a unit square.
It is easy to see that this surface is worth
By choosing , we have for all the cercle divided into equal sectors and :
because we find again the area of the unit circle. On the other hand, we have the well known series :
i.e. and and hence . The equation 5 hence becomes
This gives a family of series converging as fast as we want towards and hence improves Archimedes' algorithm .
For , if we calculate , then the serie
gives however good decimals of at each term.
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