Proving An/Bn -> L as n→∞

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In summary, the conversation is about showing that two sequences, An and Bn, converge to the same limit of 0. The speaker is questioning whether their method would allow them to replace An and An+1 with BnL and Bn+1L, respectively, in the limit as n goes to infinity, in order to show that Bn+1/Bn < 1 if An+1/An < 1.
  • #1
Fisicks
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I am given that An/Bn -> L, as n goes to infinity, where An and Bn are sequences. I also know that An and Bn converge to zero and have positive terms.
Pick some E>0, and by definition of the limit there exists an N such |An/Bn - L| < E for all n>N.
Because Bn converges to zero, the sequence is bound by M.

The next step is when my question comes in because I have never really shown that a sequence converges to a sequence.

1/M|An-BnL| < 1/Bn|An-BnL|= |An/Bn - L|< E

Therefore take E to be E/M, and An converges to BnL.

Correct?
 
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  • #2
It doesn't make sense to say that "a sequence converges to a sequence". What you are showing is that two sequences have the same limit. But if you have already said that [itex]\{A_n\}[/itex] and [itex]\{B_n\}[/itex] converge to 0, It is trivial that [itex]\{B_nL\}[/itex] also converges to 0 and so that [itex]\{A_n\}[/itex] and [itex]\{B_nL\}[/itex] converge to the same limit- 0.
 
  • #3
I should probably let you know my aim is to show that Bn+1/Bn <1 if I know that An+1/An <1. Would my method allow me to replace An+1 by Bn+1L and An by BnL in limit as n goes to infinity?
 

1. What is the meaning of "Proving An/Bn -> L as n→∞"?

The notation An/Bn -> L as n→∞ is used in mathematics to represent the limit of the ratio of two sequences An and Bn as n approaches infinity. This means that as the values of n get larger and larger, the ratio of An to Bn approaches a constant value L.

2. How is this notation used in scientific research?

This notation is commonly used in scientific research, especially in fields such as physics, chemistry, and biology. It is used to prove the convergence of certain sequences or to show the behavior of a system as it approaches infinity.

3. What is the importance of proving An/Bn -> L as n→∞?

Proving the limit of a sequence is important in understanding and predicting the behavior of a system. It allows scientists to make accurate predictions about how a system will behave as it approaches infinity, which is crucial in many areas of scientific research.

4. What are the steps involved in proving An/Bn -> L as n→∞?

The first step is to understand the definition of limit and the notation used. Then, you need to determine the limit of the numerator sequence An and the denominator sequence Bn. Next, you need to prove that the limit of the ratio An/Bn is equal to L. This can be done using mathematical proofs, such as the epsilon-delta proof method.

5. Are there any common misconceptions about proving An/Bn -> L as n→∞?

One common misconception is that the limit of the ratio An/Bn must equal L at all values of n. However, this is not always the case. The limit only needs to be equal to L as n approaches infinity, not necessarily at every value of n. Another misconception is that proving the limit is the final step in understanding a system's behavior, when in fact it is just one piece of a larger puzzle.

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