Tuesday, 15 November 2016

Binomial limit left(binom3nnbinom2nn1right)1/n as ntoinfty



The limit:limn((3nn)(2nn))1n




What I did was put limit = L. Then,



log(L)=limn1nn1r=0log(3rn2rn)=10log(3x2x)dx=log(2716)



Is this aproach correct? Is there other method.




Edit: I have corrected the expression for the limit.


Answer



limn((3nn)(2nn))1n=limn(n!(3n)!(2n)!2)1n=limnexp(ln(n!(3n)!(2n)!2)n)=limnexp(ln((2πn(ne)n)(2π(3n)(3ne)3n)(2π(2n)(2ne)2n)2)n)=limnexp(ln(36n+1224n+1)n)=limnexp(ln(e(6n+12)ln3e(4n+1)ln2)n)=limnexp(ln(2716)ln(43)2n)=2716



Solved with Stirling approximation
x!2πx(xe)x, for x


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