Re: [math-fun] Plouffe's 2 surprising approximations and 1 exact formula for pi.

25 Jan 2011


      On 1/25/2011 4:30 AM, Bill Gosper wrote:
...
Does anybody know how to coax a hundred or so digits out of
a decently convergent Sum[...]??  Total[Table[...]] is ridiculous,
and requires DIY convergence analysis.  Even in 8.0.
A little coding does it:

In[31]:= bSum[k_Real] := Module[{sum = 0, prev, n = 1, pwr, term = 1},
   pwr = Exp[2 Pi k];
   While[True,
    prev = sum;
    term *= pwr;
    sum += 1/((term - 1)*n);
    If[sum - prev == 0, Break[]];
    n++
    ];
   sum]

In[32]:= With[{k = 7`1000},
   bSum[k] -
     bSum[1/k] == (Log[k]/2 - Pi*(k - 1/k)/12)] // AbsoluteTiming

Out[32]= {0.1089891, True}

--Sasha

Re: [math-fun] Plouffe's 2 surprising approximations and 1 exact formula for pi.

Oleksandr Pavlyk