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   Limit of a Combinatorial Sum
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   Author  Topic: Limit of a Combinatorial Sum  (Read 780 times)
Michael Dagg
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Limit of a Combinatorial Sum  
« on: Aug 1st, 2008, 11:51am »
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Suppose  
 
G(m) = \sum_{i=1}^m  \sum_{j=1}^m C(m,i) C(m,j) i^{m-j} j^{m-i}  .
 
Show that
 
lim m->oo [ (G(m))^{1/(2m)}  ln m ]/m = 1/e   .
« Last Edit: Aug 3rd, 2008, 8:47am by Michael Dagg » IP Logged

Regards,
Michael Dagg
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Re: Limit of a Combinatorial Sum  
« Reply #1 on: Aug 1st, 2008, 1:54pm »
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Should that be j^{m-1} or j^{m-i}?
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Michael Dagg
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Re: Limit of a Combinatorial Sum  
« Reply #2 on: Aug 2nd, 2008, 8:38pm »
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Sorry, I made typo --   j^{m-i}   is correct.  Good observation, however, I hope  
no one spent any time on it -- as written, the sum (whose terms are all positive)  
exceeds the term when   i=j=m   and that term is   m^{m-1}  .
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Michael Dagg
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