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riddles >> putnam exam (pure math) >> Limit of a Combinatorial Sum
(Message started by: Michael Dagg on Aug 1st, 2008, 11:51am)

Title: Limit of a Combinatorial Sum
Post by Michael Dagg on Aug 1st, 2008, 11:51am
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 .

Title: Re: Limit of a Combinatorial Sum
Post by Obob on Aug 1st, 2008, 1:54pm
Should that be j^{m-1} or j^{m-i}?

Title: Re: Limit of a Combinatorial Sum
Post by Michael Dagg on Aug 2nd, 2008, 8:38pm
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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