What is `lim_(n->oo)(((n!)^2)/(n^(2n)))^(1/n)`

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justaguide eNotes educator| Certified Educator

The limit `lim_(n->oo)(((n!)^2)/(n^(2n)))^(1/n)` has to be determined.

` ` ` `n! = 1*2*3*...n < n^n which is n multiplied by itself n times.

As the denominator is greater than the numerator the term `((n!)^2)/(n^(2n))` decreases as n increases.

As n tends to inf., `((n!)^2)/(n^(2n))` tends to 0.Also, `(1/n)` tends to 0

The required limit `lim_(n->oo)(((n!)^2)/(n^(2n)))^(1/n)`

reduces to the form

`lim_(n->0) n^n` = 1

The limit `lim_(n->oo)(((n!)^2)/(n^(2n)))^(1/n)` = 1