Homework: Prove or disprove: (5n)!=O(n!^5)

Question

I have this question in my h.w:

Prove or disprove: (5n)!=O(n!^5).

I don't know how to approach this (of course I know the O notation definition but I don't have a clue how to solve it).. any help please?

Answer 1

I think we have to use Stirling's approximation here, which gives the following approximations:

Since it is known that:

We have:

If you need a more detailed proof, you can indeed use the Big O formal definition, still with Stirling's approximation, to obtain this result.