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

1 Answer
Mar 23, 2018

Answer:

#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#