Question reagarding liminf definition

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    Definition
In summary, the liminf of a sequence is the least upper bound of all subsequential limits and is not necessarily smaller or larger than any individual term in the sequence.
  • #1
transgalactic
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regarding this definition

[itex]

a = \lim \bigg( \inf \{ a_k | k\geq n\} \bigg)
[/itex]

the sequence
[itex]

\inf \{ a_k | k\geq n \}
[/itex]

is non decreasing . its inf gets bigger or not changing in each following sequence
so its limit is its least upper bound

am i correct??
 
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  • #2
Yes.
 
  • #3
what is the relations between this liminf and
all the members in the sequence

is it bigger or smaller then all of them?
 
  • #4
There is no requirement that a "liminf" be smaller or larger than all members of the sequence. The liminf of a sequence is the least upper bound of all subsequential limits.

For example, suppose [itex]a_n[/itex] is (n-6)/2n if n is odd, -(n-6)/2n if n is even. Then {[itex]a_n[/itex]} is {-5/2, 1, -1/2, 1/4, -1/10, 0, 1/14, -1/8, ...}. For n odd, we have a sequence that converges to 1/2. For n even, we have a sequence that converges to -1/2. The liminf is the smaller of those, -1/2 but there is a number in the sequence less than -1/2. The limsup is 1/2 but there is a term of the sequence larger than 1/2. We can change any finite number of terms in a sequence with changing any subsequential limits so there cannot be any relation between the limit or liminf or limsup and individual terms of the sequence.
 

1. What is the definition of liminf?

The liminf (limit inferior) of a sequence is the smallest number that the elements of the sequence approach as the index approaches infinity. It is denoted as liminfn→∞ or limn→∞ inf n.

2. How is liminf different from limsup?

Liminf and limsup (limit superior) are two types of limits for a sequence. While liminf is the smallest limit that the elements of a sequence approach, limsup is the largest limit that the elements approach. Liminf represents the "lower bound" of a sequence, while limsup represents the "upper bound."

3. Can liminf be equal to limsup?

Yes, liminf and limsup can be equal in certain cases. For example, if the sequence is a constant sequence (where all elements are the same), then liminf and limsup will be equal. However, in most cases, liminf and limsup will be different.

4. What is an example of calculating liminf?

Consider the sequence {1, -2, 3, -4, 5, -6, ...}. The liminf of this sequence is -∞ because as the index approaches infinity, the elements of the sequence will alternate between approaching positive and negative infinity. Therefore, the smallest limit that the elements approach is -∞.

5. How is liminf used in mathematical analysis?

Liminf is used in mathematical analysis to determine the convergence or divergence of a sequence. If the liminf of a sequence is a finite number, then the sequence is convergent. If the liminf is infinity, then the sequence is divergent. Additionally, liminf can be used to find the infimum (greatest lower bound) of a set of real numbers.

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