Thursday, 12 December 2013

calculus - How to prove that K=limlimitsntoinftyleft(prodlimitsnk=1akright)1/napprox2.6854520010?

I was going through a list of important Mathematical Constants, when I saw the Khinchin's constant.



It said that :




If a real number r is written as a simple continued fraction :




r=a0+1a1+1a2+1a3+, where ak are natural numbers k, then lim exists and is a constant K \approx 2.6854520010, except for a set of measure 0.




First obvious question is that why the value a_0 is not included in the Geometric Mean? I tried playing around with terms and juggling them but was unable to compute the limit. Also, is it necessary for r to be "written-able" in the form of a continued fraction ?



Thanks in Advance ! :-)

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