Limit Inferior and Limit Superior

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In summary, the conversation discusses the concept of limit superior and limit inferior for a positive sequence of real numbers and the relationship between them. It is shown that for any positive sequence, the inequality lim inf (a _(n+1) / a_n) <= lim inf (a_n)^(1/n) <= lim sup (a_n)^(1/n) <= lim sup(a_(n+1) / a_n) holds. Examples are provided to illustrate where equality does not hold. The importance of understanding the definitions of inf, sup, lim inf, and lim sup is emphasized.
  • #1
veronicak5678
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Prove that for any positive sequence a_n of real numbers
lim inf (a _(n+1) / a_n) <= lim inf (a_n)^(1/n) <= lim sup (a_n)^(1/n)
<= lim sup(a_(n+1) / a_n).
Give examples where equality does not hold.

Is lim sup always >= lim inf? I am having trouble understanding these concept and proving things about them without specific numbers.
 
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  • #2
For your second question, the answer is obviously yes. Inf is always ≤ sup, so this relationship will hold as you take limits.

An example for the first question, the following sequence will work:
an = 2n for even n.
an = 1/2 for odd n.

lim inf an+1/an = 0
lim inf an1/n = 1
lim sup an1/n = 2
lim sup an+1/an = ∞
 
  • #3
Could you please show how you got those values? I am obviously confused about the definition. My book isn't making much sense to me.
 
  • #4
I suggest you get a better handle on the meanings of inf, sup, lim inf, and lim sup.

For the ratio, when n is odd the value is 2n+2, and when n is even the value is 2-n-1, so as n becomes infinite, the lim inf = 0 and the lim sup is infinite.
 
  • #5
http://en.wikipedia.org/wiki/Limit_superior_and_limit_inferior

The graphic on this page is a useful illustration. (Wikipedia has an unfortunate habit of getting extremely technical, and not separating the complex concepts form the simpler ones, so if the rest of the page isn't all that helpful, don't be discouraged.)

Basically, (and these definitions are not rigorous) the lim sup is the is the "smallest value that the sequence is *eventually* not bigger than" and the lim inf is the "largest value that the sequence is eventually not smaller than." Compare these descriptions with the picture, and hopefully it'll make more sense.
 

1. What is the definition of limit inferior and limit superior?

Limit inferior and limit superior are concepts in real analysis that describe the behavior of a sequence or a function as the input approaches a certain value, usually infinity. They represent the smallest and largest possible limit points of the sequence or function, respectively.

2. How are limit inferior and limit superior related to each other?

Limit inferior and limit superior are related in that they both provide information about the limiting behavior of a sequence or function. While limit inferior represents the smallest possible limit point, limit superior represents the largest possible limit point. They are considered to be complementary concepts as they provide bounds on the possible limits of a sequence or function.

3. Can limit inferior and limit superior be equal?

Yes, it is possible for limit inferior and limit superior to be equal. This occurs when the sequence or function has a unique limit point. In this case, the limit inferior and limit superior will both converge to the same value.

4. How do limit inferior and limit superior relate to the concept of limit?

Limit inferior and limit superior are closely related to the concept of limit. While limit represents the actual value that a sequence or function approaches as the input approaches a certain value, limit inferior and limit superior represent the bounds on the possible limit points. In other words, limit inferior and limit superior can be seen as a range within which the limit of a sequence or function must fall.

5. What are some real-world applications of limit inferior and limit superior?

Limit inferior and limit superior have various applications in mathematics, physics, and engineering. They are used to determine the stability and convergence of sequences and functions, as well as to analyze the behavior of systems and processes. For example, in economics, limit inferior and limit superior can be used to study the behavior of stock prices over time, while in physics, they are used to analyze the behavior of waves and oscillations.

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