Linear convergence also for the one on the first page of the pdf (and to 2*Pi, not Pi), using Pari/gp: u=0.5; fa(n, u)=2^n*sqrt(1-u*u)*(2+1/u); { for (n=1, 20, a = fa(n, u); print([n, a]); u = sqrt(2*u + 2)/2; ); } [ 1, 6.928203230275509174109785366023489467771221015241522512] [ 2, 6.309401076758503058036595122007829822590407005080507504] [ 3, 6.284698261089313849362810669937794318042660390635843197] [ 4, 6.283278112439984585093422611211121983684728189547233237] [ 5, 6.283191080816779590245594957769397689710707177378823561] [ 6, 6.283185667617591716261043297794296056715862349224554543] [ 7, 6.283185329700498681764470474408453804919470421723461451] [ 8, 6.283185308587042544838278336102423311351107299137065124] [ 9, 6.283185307267550904067606378734738531926108069644277105] [10, 6.283185307185084228980090604450773526344097135976414163] [11, 6.283185307179930086043689611234615188055755351473937852] [12, 6.283185307179607952489170979060409544946653401185434692] [13, 6.283185307179587819147935530420307409300710608427640595] [14, 6.283185307179586560814200845559546060792694428128468339] [15, 6.283185307179586482168343873547465250587906278534678652] [16, 6.283185307179586477252977835387205202554516100676935709] [17, 6.283185307179586476945767458355165431733785104475147206] [18, 6.283185307179586476926566809796178203582845220233949936] [19, 6.283185307179586476925366769261327677722211068633393702] [20, 6.283185307179586476925291766727900866354340044649732273] Next! Best regards, jj * Simon Plouffe <simon.plouffe@gmail.com> [Mar 27. 2017 18:51]:
Hello,
I programmed the first one,
is is not like the AGM, the number of valid digits is proportional to n, something like approx. (5/2)^n,
here is the maple routine :
# F.M. Essomba formula
es:=proc(n) option remember; if n = 0 then 1/2 elif n = 1 then sqrt(3.0)/2 else sqrt(2.0*es(n-1) + 2)/2.0 end if: end;
a:=proc(n) 3*2.0^n/315*sqrt(1.0-es(n)^2)*(256 + 597*es(n) - 1024*es(n)^2 + 542*es(n)^3-56*es(n)^5);end;
a(10000) differs from Pi at the 3979'th position. a (4000) differs from Pi at the 1591'th position,
confirming that it converges rather slowly.
It is not a record calculation, but certainly original (IMHO).
I have not tried the other ones yet.
Have a nice day and bonne journée.
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