Ratio Test - Lim sup and lim inf version -

Your name]In summary, the lim sup and lim_{n -> \infty} are both tests for convergence, but they differ in their approach. The lim sup is the supremum of all possible subsequential limits, while the lim_{n -> \infty} is simply the limit of the sequence as n approaches infinity. In some cases, the lim sup may provide a more accurate representation of the convergence behavior of the sequence.
  • #1
michonamona
122
0
Hello friends,

Homework Statement


[tex] lim sup |\frac{a_{n+1}}{a_{n}}|[/tex]

I know that this operation tests for convergence, but I don't understand how it's related to [tex] lim_{n -> \infty}|\frac{a_{n+1}}{a_{n}}|[/tex]

I understand that both will give the same result, but how is the latter different? lim sup is the sup of the collection of sub-sequential limits, so what does it mean when we take [tex] lim sup |\frac{a_{n+1}}{a_{n}}|[/tex] ?I appreciate your help,

M
 
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  • #2
aggie

Dear Maggie,

Thank you for your question. You are correct in saying that both the lim sup and lim_{n -> \infty} are tests for convergence. However, they are slightly different in their approach.

The lim sup, or limit superior, is the supremum (or least upper bound) of the set of all possible subsequential limits of a sequence. In other words, it is the largest value that a subsequence of the original sequence can converge to. This is why we take the supremum, rather than just the limit of the sequence itself. It allows us to consider all possible subsequences and their limits.

On the other hand, the lim_{n -> \infty} is simply the limit of the sequence as n approaches infinity. This means that we are only considering the behavior of the original sequence itself, rather than any possible subsequences.

In the case of |\frac{a_{n+1}}{a_{n}}|, both the lim sup and lim_{n -> \infty} will give the same result because the sequence is bounded and therefore all its subsequences will converge to the same limit. However, in more complex cases, the lim sup may provide a more accurate representation of the convergence behavior of the sequence.

I hope this helps clarify the difference between the lim sup and lim_{n -> \infty}. Let me know if you have any further questions.
 

What is the purpose of the Ratio Test - Lim sup and lim inf version?

The Ratio Test - Lim sup and lim inf version is a mathematical test used to determine the convergence or divergence of a series. It is particularly useful for series with alternating signs or terms that are difficult to evaluate directly.

How does the Ratio Test - Lim sup and lim inf version work?

The test involves taking the limit of the ratio of consecutive terms in a series. If the resulting limit is less than 1, the series is guaranteed to converge. If the limit is greater than 1, the series diverges. If the limit is equal to 1, the test is inconclusive and another test must be used.

What is the difference between lim sup and lim inf in the Ratio Test?

Lim sup (limit superior) and lim inf (limit inferior) are two different ways of approaching the limit in the Ratio Test. Lim sup looks at the largest possible limit that can be obtained by taking the limit of subsequences of the original series, while lim inf looks at the smallest possible limit. In some cases, only one of these limits will exist, while in others both will exist and give different values.

What are the conditions for the Ratio Test - Lim sup and lim inf version to be applicable?

The series must have positive terms and the limit of the ratio of consecutive terms must exist. Additionally, for the test to be conclusive, the limit must be less than 1 or greater than 1. If the limit is equal to 1, another test must be used to determine the convergence or divergence of the series.

Are there any drawbacks to using the Ratio Test - Lim sup and lim inf version?

While the Ratio Test is a useful tool for determining convergence or divergence of a series, it is not applicable to all series and can sometimes give inconclusive results. It is important to understand and consider other tests as well when analyzing the convergence or divergence of a series.

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