The discussion revolves around the limit of a sequence where the limit L is greater than 1. Participants are exploring different approaches to demonstrate or prove the behavior of the sequence as it approaches infinity.
The discussion is active, with participants sharing their thoughts on the proof methods and the implications of the limit. Some have offered insights into the sequence's behavior, while others are questioning assumptions and exploring different interpretations of the problem.
There is a mention of a book that provides a proof for when L is less than 1, which contrasts with the current scenario of L being greater than 1. This highlights the need for clarity on the conditions under which the proofs apply.
SammyS said:I doubt that he wants an ε - N proof.
Probably wants more than a rough demonstrative "proof". I would.
Do a comparison test. The hint is right there !
I It doesn't say xn = (sn) at all.Shackleford said:xn = (sn)
sn+1/sn = xn+1/xn = x*xn/xn = x
x ≥ L > 1
Eh. I think that "proves" it. Assuming x > 1, then the sequence ratio is bigger than 1. And, of course, the limit is given which is infinity which is what we were asked to show for a sequence that satisfies that ratio.
SammyS said:I It doesn't say xn = (sn) at all.
sn+1 ≥ L sn ≥ L2 sn-1 ≥ L3 sn-2 ≥ ...
What can you say about \displaystyle \lim_{n\to\infty}L^n\,?
This leads nicely to an ε - N proof. Maybe do one.
Shackleford said:That limit should go to zero, right?
The book does the proof for when L < 1. I'm looking over that proof now.
SammyS said:No, it doesn't go to zero. L > 1 !