Is the below nested radical converge?

Question

Sorry i can't write the below nested radical using latex , I accrossed it in some web , i w'd like if it is a convergent or no ? and thanks

Answer 1

Your expression can be written as $$ 2^{1/2!} 3^{1/3!} 4^{1/4!} \ldots = \prod_{n=2}^\infty n^{1/n!} = \prod_{n=2}^\infty \exp\left(\frac{\log n}{n!}\right) = \exp\left(\sum_{n=2}^\infty \frac{\log n}{n!}\right)$$ The sum is easily seen to converge, and therefore the product converges as well.