For a strictly positive sequence, prove that .
I get that it's trying to say that the root test converges faster than the ratio test (hence the greater limit inf), but beyond that, I am drawing a blank. I don't even know where to begin.
For a strictly positive sequence, prove that .
I get that it's trying to say that the root test converges faster than the ratio test (hence the greater limit inf), but beyond that, I am drawing a blank. I don't even know where to begin.
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)
Infinum means greatest lower bound. I.e., if A = (0,1), inf A = 0, because 0 <= all elements in A, and 0 is the greatest possible number that satisfies that inequality.
Lim inf x_n = lim (n -> infinity) {a_n, a_n+1, a_n+2, ...}. Basically, lim inf is the infinum of the sequence infinitely far from the starting element.
All of that I get. What I don't get is how to put it together in a meaningful way on this proof.