Saturday, 29 October 2016

real analysis - Computing: limlimitsntoinftyleft(prodlimitsnk=1binomnkright)frac1n



I try to compute the following limit:




\lim_{n\to\infty}\left(\prod_{k=1}^{n} \binom{n}{k}\right)^\frac{1}{n}



I'm interested in finding some reasonable ways of solving the limit. I don't find any easy approach. Any hint/suggestion is very welcome.


Answer



All the binomial coefficients except the last one are at least n, so the nth root is at least n^{\frac{n-1}{n}}, so the limit is infinity.


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