It's amazing that Khinchin found that for almost all positive reals x, the GM of the convergents of its CF expansion are independent of x. Is there a simple reason why this should be true? --Dan On 2012-12-24, at 5:23 AM, Hans Havermann wrote:
Under 'Applications' here < http://reference.wolfram.com/mathematica/ref/Khinchin.html >, the Mathematica documentation states: Geometric mean of the first 1000 continued fraction terms in Pi…
In[1]:= N[Apply[Times,ContinuedFraction[Pi,1000]]^(1/1000)] Out[1]:= 2.66563
In[2]:= N[Khinchin] Out[2]:= 2.68545
Sadly, the example ignores the fact that the first term of a continued fraction is very different from all the rest, a matter that would have been more apparent if one had asked for the geometric mean of the first 1000 continued fraction terms of Pi/4.
I have just determined that the geometric mean of 1498931686 terms (cherry-picked) of the *fractional* part of the continued fraction of pi is within 1.002405*10^-13 of Khinchin's constant.
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