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Boris Gourévitch
The world of Pi - V2.57
modif. 13/04/2013



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8 And the other bases then ? ?

The formulae of the previous section are applicable at all bases 2k  and 3k  . Good. But it would be great if we could access representation of our favourite constant in other bases !

Unfortunatly, this is for the moment not directly possible with BBP formulae. This is due to the fact that the integrals complicated a lot the task in base 5, 7 or other, by giving some uncomon logarithm or arctan.

Hence, we have for example

 oo            integral 
 sum  1---1--=   1--5---dy = - 1 V~ 5-ln 2+ 1  V~ 5ln(3 +  V~ 5)
i=0 5i2i+ 1   0 5 - y2      2        2
(305)

                                      (    )
 oo  sum  (- 1)i  1     integral  1 7        1 V~          V~ 7
   -7i--2i+-1 =   7-+-y2dy = 2 7 arctan  -3-
i=0              0
(306)

However, we can find certain interesting combination such as this formula by Gery Huvent

 oo  sum         i      sum  oo     (             )     V~ -
   ---(-1)----+     -1-- --4-- - --1--  = p--6
i=06i+1(2i+ 1)  i=0 8i+1  6i+ 1   6i+ 5     12
(Huvent)


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