Proving the Convergence of An->1: A Debate on An+1/An Ratio

  • Thread starter transgalactic
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    Convergence
In summary: Now, what do you know about the size of A[sub]nA_{n+1] and 1?In summary, the conversation discusses how to prove or disprove the statement that if An converges to 1, then An+1/An also converges to 1. The conversation also explores the concept of monotone sequences and the use of limits in proofs. The final question raises the topic of disproving a statement, specifically in the case of (An)^n ->1.
  • #1
transgalactic
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prove or desprove that if An->1


then An+1/An ->1


??
 
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  • #2
Anytime the question says "prove or disprove" and you're not sure what to do, start by spending five minutes trying to generate a counterexample. If you can't do it, see if you can figure out what's stopping you (which usually leads to a proof that the statement is true)
 
  • #3
i know that if a series is converges and monotone
then if An->1 then An+1 ->1

so i construct a limit An+1/An
n-> +infinity
and say that if both members go to 1 then the whole expression goes to 1

the problem is:
i don't know if An is monotone
and if this way is correct and will be regarded as a formal proof
??
 
  • #4
transgalactic said:
i know that if a series is converges and monotone
then if An->1 then An+1 ->1

You don't need the monotone condition here. For all e>0, there exists N>0 such that n>N implies |An - 1| < e gives us that |An+1 - 1| < e for n>N also as n+1>n>N
 
  • #5
and after writing what you said
i do the limit part?
 
  • #6
Yeah
 
  • #7
thanks
 
  • #8
You have been posting questions here long enough to have learned to copy the problem correctly even if you do not wish to use Latex.

Is this A_{n+1}/A_n or (A_n+ 1)/A_n ?
 
  • #10
its A(n+1)/A(n)
 
  • #11
when i solve this type of question:
how do i recognize that the expression doesn't converge
and how do i disprove that the function converges
in such case?
 
  • #12
i need to prove that (An)^n ->1

but when i construct limit
lim (An)^n
n->+infinity

i get 1^(+infinity) which says that there is no limit
what do i do in this case in order to disprove that (An)^n->1

??
 
  • #13
Why do you need to prove that Ann goes to 1? Is this a completely different question?

As to your original question, if An converges to 1, then, given any [itex]\epsilon> 0[/itex] there exist N such that if n> N, [itex]1-\epsilon\le A_n\le 1+\epsilon[/itex]. If n> N, that's true for both An and An+1.
 

1. What is the purpose of proving the convergence of An->1?

The purpose of proving the convergence of An->1 is to determine whether a sequence, An, approaches a limit of 1 as n approaches infinity. This is important in mathematical analysis and can help us understand the behavior of various functions.

2. How is the convergence of An->1 typically proven?

The convergence of An->1 is typically proven using mathematical techniques such as the Ratio Test, the Limit Comparison Test, or the Root Test. These tests involve evaluating the ratio or limit of the sequence and comparing it to known values to determine if the sequence approaches a limit of 1.

3. What is the importance of the An+1/An ratio in this debate?

The An+1/An ratio is important in proving the convergence of An->1 as it is used in various mathematical tests to determine the behavior of the sequence. By evaluating this ratio, we can determine whether the sequence approaches a limit of 1 or diverges.

4. Can the convergence of An->1 be proven for all types of sequences?

No, the convergence of An->1 cannot be proven for all types of sequences. Some sequences may not have a limit of 1, while others may have a limit that is difficult to determine. In these cases, alternative methods may be used to analyze the behavior of the sequence.

5. How does proving the convergence of An->1 impact the overall understanding of a mathematical concept?

Proving the convergence of An->1 can greatly impact the overall understanding of a mathematical concept. It allows us to determine the behavior of a sequence, which can in turn help us understand the behavior of a function, series, or other mathematical concept. It also provides a basis for further analysis and applications in various fields of science and engineering.

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