$lim_(n->oo)(1/n((n+1)(n+2)(n+3)cdots(2n))^(1/n))?$

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Answer 1 · 2016-12-13T17:14:17

$4/e$

we have

$((n+1)(n+2)(n+3)cdots(2n)) =((2n)!)/(n!)$

$log_e(n!) approx nlog_en-n$ in

$y = 1/n(((2n)!)/(n!))^(1/n)$

we have

$y = 1/n(((2n)^(2n)e^(-2n))/(n^n e^(-n)))^(1/n)=4/e$

So the answer is

$lim_(n->oo)(1/n((n+1)(n+2)(n+3)cdots(2n))^(1/n))=4/e$

lim_(n->oo)(1/n((n+1)(n+2)(n+3)cdots(2n))^(1/n))?lim_(n->oo)(1/n((n+1)(n+2)(n+3)cdots(2n))^(1/n))?