Question

Limit (n~■)[2n!÷n!n^n]^(1÷n) here■is infinitive?

Answer 1

`4/e`

Explanation:

#using the Stirling asymptotic formula

`n! approx sqrt(2pi n)(n/e)^n`

`(((2n)!)/((n!)n^n))^(1/n) approx 1/n (sqrt(2pi(2n))/sqrt(2pi n))^(1/n) ((2n)/e)^2/(n/e) = 1/n 2^(2/n)(4n)/e = 2^(2/n)4/e`

hence

`lim_(n->oo)(((2n)!)/((n!)n^n))^(1/n) = lim_(n->oo) 2^(2/n)4/e = 4/e`