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 [itex]\displaystyle \lim_{n\to\infty}L^n\,?[/itex]
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 !
A sequence limit proof is a mathematical technique used to show that a sequence of numbers converges to a specific value, known as the limit. It involves using logical arguments and mathematical equations to demonstrate that the terms of the sequence get closer and closer to the limit value as the sequence progresses.
Proving the limit of a sequence is important because it allows us to understand the behavior of the sequence as it approaches the limit. This information can be used to make predictions and draw conclusions in various fields such as physics, engineering, and economics.
The steps involved in a sequence limit proof typically include defining the limit value, showing that the sequence is bounded, and using the definition of the limit to show that the sequence approaches the limit value. This may also involve using other mathematical techniques such as mathematical induction or the epsilon-delta method.
Some common challenges in sequence limit proofs include dealing with complex sequences, choosing the appropriate mathematical techniques, and avoiding logical fallacies. It is also important to double-check all steps and assumptions made in the proof to ensure its validity.
Yes, sequence limit proofs have various real-life applications. For example, they are used in physics to study the behavior of systems that approach a steady state, in economics to analyze the long-term behavior of financial markets, and in computer science to analyze the performance of algorithms as the input size increases.