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.

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- Apr 21st 2013, 06:16 PMphys251Limit infina of ratio test and root test
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. - Apr 22nd 2013, 12:16 AMGusbobRe: Limit infina of ratio test and root test
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:

Quote:

I get that it's trying to say that the root test converges faster than the ratio test (hence the greater limit inf)

- Apr 22nd 2013, 02:17 PMphys251Re: Limit infina of ratio test and root test
I still do not understand. At all.

- Apr 22nd 2013, 09:17 PMGusbobRe: Limit infina of ratio test and root test
Do you understand the definition of liminf? Can you see that if a sequence a_n converges, liminf a_n = lim a_n = limsup a_n?

- Apr 23rd 2013, 07:35 AMphys251Re: Limit infina of ratio test and root test
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. - Apr 23rd 2013, 03:08 PMGusbobRe: Limit infina of ratio test and root test
If the sequence is not eventually constant, there are infinitely many points where . Then . The case where is eventually constant is obvious.