Question

Doing series convergent tests, and on a roots test, I get to a limit that I don't know how to prove.

Prove:
Limit n to infinity of ((n!)^5)/((5n)!) = 0
Do i need to do L'Hospitals rule? If so, how do I take the derivative of the top and bottom.

Answer 1

\[\lim_{n \rightarrow \infty} \frac{ n!^5 }{ (5n)! }\]

Answer 2

How did you come about this limit?

Answer 3

\[\sum_{n=1}^{\infty} \frac{ n!^{5n} }{ (5n)!^{n} }\]
This is the origional sum, when doing the roots test, you factor out n, take the nth root, and take the limit of the absolute value of a sub n.

Answer 4

I would say that this series does not converge.

Answer 5

Did u try d'Alemberts ratioj test?