I did a numerical experiment in which I rounded the denominators to the nearest 4n+1, and got pi times something close to the continued fraction 0/1A12222... (The A is a 10), which simplifies to (219+V2)/241. (I started with 3, not 5, figuring that the extra factor of 3/1 would just come out in the wash.) When I went to denominators of the form 4n+3, however, things looked a bit murkier. I went up to 10^6 and then 10^7, and the partial product was pi times 0/11152... That's as far as the continued fractions agreed. I'm peering at denominators of the form 5n, and they don't look very promising either. Now I'm starting to think that run of 2's in the cf for the 4n+1 case was a coincidence. On Fri, Feb 18, 2011 at 11:53 AM, Warut Roonguthai <warut822@gmail.com>wrote:
Changing the denominators to the closest multiple of 6, and we have
pi*sqrt(3)/6 = (5/6) (7/6) (11/12) (13/12) (17/18) (19/18) (23/24) ...
Warut
On Mon, Feb 14, 2011 at 4:09 PM, Christian Boyer <cboyer@club-internet.fr> wrote:
In www.mathpages.com/home/kmath477.htm, explanation of:
3 5 7 11 13 17 19 π/4 = - × - × - × -- × -- × -- × -- × ... 4 4 8 12 12 16 20
Here the numerators are the odd primes, and the denominators are the closest numbers of the form 4n.
Christian.
-----Message d'origine----- De : math-fun-bounces@mailman.xmission.com [mailto: math-fun-bounces@mailman.xmission.com] De la part de Bill Gosper Envoyé : lundi 14 février 2011 08:24 À : math-fun@mailman.xmission.com Objet : Re: [math-fun] Euler's crazy pi product
interesting how if you start with 1 you get a line of slope 1. you can also include all integers.
π/4 = 1/2 × 2/2 × 3/2 × 4/4 × 5/6 × 6/6 × 7/6 ...
if I've got that right.
For the n-1/2 cases, round up and round down produce divergent products. Round to even
gives 4*(1/4)!^2/sqrt(2*pi) . At last, a mathematical justification for round to even! --rwg
On Sun, Feb 13, 2011 at 9:36 PM, Bill Gosper <billgosper@gmail.com < http://gosper.org/webmail/src/compose.php?send_to=billgosper%40gmail.com>> wrote:
http://math.ucr.edu/home/baez/week127.html finds in Lennart Berggren, Jonathan Borwein and Peter Borwein, π: A Source Book, Springer-Verlag, New York, 1997, "the following weirdly beautiful formula due to Euler, which unfortunately is not explained:" (Then how can they call it a sourcebook?--rwg)
3 5 7 11 13 17 19 π/2 = - × - × - × -- × -- × -- × -- × ... 2 6 6 10 14 18 18
"Here the numerators are the odd primes, and the denominators are the closest numbers of the form 4n+2." E.g., for 10, 100, ..., 10^6 terms, In[209]:=
Table[Block[{p=2},Nest[N[#*(p=NextPrime[p])/(2+4*Round[(p-2)/4]),9]&,2 ,10^n]],{n,6}]
Out[209]= {3.10152045, 3.13398462, 3.13772561, 3.14073685, 3.14143290, 3.14157195}
Why the heck?? Was Euler a preincarnation of Ramanujan? --rwg
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