Hint: For , . In particular, .
Now think about why . (It may be useful to also think about why ). In particular, what happens if the sequence is eventually constant? What if it never converges?
As an aside, this:
is not the way to think about it. I'm not even sure if makes sense. Anyways, the following is true for :I get that it's trying to say that the root test converges faster than the ratio test (hence the greater limit inf)