Limit of ((3n)!/(n!)^3)^1/n as n tends to infinity

We need to find the limit of the expression ((3n)!/(n!)^3)^1/n as n tends to infinity.

Using Stirling's approximation

We can use Stirling's approximation to simplify the expression ((3n)!/(n!)^3)^1/n. Stirling's approximation states that n! is approximately equal to (n/e)^n * sqrt(2*pi*n) as n tends to infinity.

Using Stirling's approximation, we can rewrite the expression ((3n)!/(n!)^3)^1/n as:

((3n/e^(3n))*(2*pi*3n)^(1/2))/((n/e^n)^3)

Simplifying this expression, we get:

(3^(3/2)*2*pi)^(1/2)*e^(2n/3)/n^(3/2)

Limit as n tends to infinity

Taking the limit of the above expression as n tends to infinity, we get:

lim (n->inf) ((3n)!/(n!)^3)^1/n = lim (n->inf) (3^(3/2)*2*pi)^(1/2)*e^(2n/3)/n^(3/2) = 0

Therefore, the limit of ((3n)!/(n!)^3)^1/n as n tends to infinity is 0.


Note: The above explanation assumes that n is a positive integer. If n is a real number, then the limit does not exist.