Another limsup and lim inf question

  • Thread starter Ed Quanta
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In summary, the limsup (limit superior) and lim inf (limit inferior) are important concepts in mathematical analysis used to describe the behavior of a sequence of numbers or functions. They are closely related to the supremum and infimum and can be used to prove the convergence of a sequence. They can also be calculated for infinite sequences and have practical applications in various fields.
  • #1
Ed Quanta
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So suppose an approaches L,

then how do I know that the statement

1)If lim supbn<b, then limsup(anbn)<Lb

implies that

limsup(anbn)<=L*limsup(bn)

is true?


2)How do I know that

limsup(anbn)=L*limsup(bn)

implies

liminf(anbn)=L*liminf(bn)

is true?
 
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  • #2
i suggest you go back and read what some of us said in answer to your previous post. you want to get some skill with this idea yourself. the methods are the same.
 
  • #3
Thanks dude. I'll look back.
 

1. What is the definition of limsup and lim inf in mathematics?

The limsup (limit superior) and lim inf (limit inferior) are concepts used in mathematical analysis to describe the behavior of a sequence of numbers or functions. The limsup is the largest limit point of the sequence, while the lim inf is the smallest limit point.

2. How are limsup and lim inf related to the concepts of supremum and infimum?

The limsup and lim inf are closely related to the supremum (least upper bound) and infimum (greatest lower bound) of a sequence. In fact, the limsup is the supremum of the set of all limit points of the sequence, while the lim inf is the infimum of the same set.

3. How are limsup and lim inf used in proving the convergence of a sequence?

When proving the convergence of a sequence, we can use the limsup and lim inf to show that the sequence has a finite limit. If the limsup and lim inf are equal, then the sequence converges to that common value. If they are different, the sequence diverges.

4. Can limsup and lim inf be calculated for infinite sequences?

Yes, limsup and lim inf can be calculated for infinite sequences. However, they may not always be finite. If the sequence is unbounded, then the limsup will be infinite, and if the sequence is decreasing without a lower bound, then the lim inf will be negative infinity.

5. Are there any practical applications of limsup and lim inf?

Yes, limsup and lim inf have many practical applications in fields such as physics, engineering, and economics. They are used to analyze the behavior of complex systems and to make predictions about their future behavior. For example, they can be used to study the growth rate of populations or the stability of a financial market.

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