Understanding Sup of Sequence: S_n

  • Thread starter Bachelier
  • Start date
  • Tags
    Sequence
In summary, Sup is the least of all upper bounds. Sup of a sequence is simply the smallest number that is still greater than every single element. It is a way to define something like max, but for all sequences.
  • #1
Bachelier
376
0
Is my understanding of the concept of ##\underset{n}{Sup} \ S_n## correct?

for instance, given the sequence:

##{S_n} = sin(\frac{n \pi}{2}). \frac{n+2}{2 n}##

Then

##\underset{1}{Sup} \ S_n \ = \ \frac{3}{2}##

##\underset{10}{Sup} \ S_n \ = 0##

##\underset{k≥n}{Sup} \ S_n \ = \ \frac{1}{2}##

I am trying to understand the part when we say ##\underset{n}{Sup} \ S_n##, what does it mean? Thanks
 
Physics news on Phys.org
  • #2
I doesn't look like you really understand sup. The n below sup is meant as a variable, and sometimes we have restrictions such as
[tex]\sup_{n\geq 5} x_n[/tex]
which means we consider sup of the sequence [itex]x_5,x_6,x_7,\ldots[/itex]. Therefore your first two statements do not seem to make sense (at least with standard notation). One could argue that formally we should have
[tex]\sup_{n=1} S_n = S_1 = 3/2[/tex]
but I have never seen anyone use sup for a single value since it is always equal to just the value.

To understand sup you should first understand max of a sequence. Some sequences have a maximum value (i.e. you can find a fixed k such that [itex]x_k \geq x_n[/itex] for all n). When this is the case then
[tex]\sup_{n} x_n = \max_n x_n = x_k[/tex]
However max does not always make sense. For instance consider the sequence
[tex](0, 1/2, 2/3, 3/4, \ldots)[/tex]
This sequence does not have a maximum because every element gets bigger. However we can see that the entries get arbitrarily close to 1. This is precisely what sup is. sup of a sequence is simply the smallest number that is still greater than every single element. It is a way to define something like max, but for all sequences.

The sequence you have given actually has a maximum value so the sup is just that value. I'm not going to point out what the value is because you should be able to see it yourself and in case this is homework. I can at least tell you that it is not 1/2, you can find elements that are greater than 1/2.
 
  • #3
No Sup for you! NEXT.
 
  • #4
Bachelier said:
Is my understanding of the concept of ##\underset{n}{Sup} \ S_n## correct?

for instance, given the sequence:

##{S_n} = sin(\frac{n \pi}{2}). \frac{n+2}{2 n}##

Then

##\underset{1}{Sup} \ S_n \ = \ \frac{3}{2}##

##\underset{10}{Sup} \ S_n \ = 0##

##\underset{k≥n}{Sup} \ S_n \ = \ \frac{1}{2}##

I am trying to understand the part when we say ##\underset{n}{Sup} \ S_n##, what does it mean? Thanks

Sup is also called least-upper bound. The sup is the least of all upper bounds. A simple

example : take the interval (0,1) --take it, please!. No, sorry, now, what are the

possible upper bounds for the set of all x's in (0,1)? Well, 2 is an upper bound, so is 3,

and so is any number larger than 3. But which is the least among all upper bounds?

It is 1. It is a little involved, but not too hard to show this.

Now, you need to do the same for your collection of objects Sn . Notice, Sn is

the set {S1,S2,...} . Out of all the numerical values of Sn, can you think of the

least real number that is larger than all the Sn's? If you can figure out, a proof

should not be too far.
Sn
 
  • #5
rasmhop said:
The sequence you have given actually has a maximum value so the sup is just that value. I'm not going to point out what the value is because you should be able to see it yourself and in case this is homework. I can at least tell you that it is not 1/2, you can find elements that are greater than 1/2.

Thank you ramhop. It is not homework. I took the example from a youtube video (see link below)

(Warning: Sie haben Deutsch zu sprechen) :)

https://www.youtube.com/watch?v=bbuYHvTVDio​
I understand the Least Upper Bound property, but what confuses me is the "limsup" notation. I kind of understand it as the limit of the sup of the tail of a certain series.
i.e. ##T_n = \left\{{S_k | k ≥ n}\right\}## for some sequence ##(S_n)##
Please feel free to expand on this!

The examples I posted were a meek attempt to try and understand some notations I found in the wikipedia article about the subject on

http://en.wikipedia.org/wiki/Limit_superior_and_limit_inferior

for instance see the image below:

Lim_sup_example_5.png

especially the inequality:## \underset{n}{Inf} \ S_n \ ≤ \underset{n→∞}{limInf} \ S_n ≤ \underset{n→∞}{limSup} \ S_n ≤ \underset{n}{Sup} \ S_n##

Now how do I look at: ##\underset{n}{Sup} \ S_n## ? should it be considered as the sup at a certain n or as the sup of all ##S_k## s.t. ##k ≥ n## or the sup of the whole sequence? I think the correct answer is the last one.
 
Last edited:
  • #6
Bacle2 said:
No Sup for you! NEXT.

Good one. :tongue: I prefer Salads anyway.
 
  • #7
Bacle2 said:
Sup is also called least-upper bound. The sup is the least of all upper bounds. A simple

example : take the interval (0,1) --take it, please!. No, sorry, now, what are the

possible upper bounds for the set of all x's in (0,1)? Well, 2 is an upper bound, so is 3,

and so is any number larger than 3. But which is the least among all upper bounds?

It is 1. It is a little involved, but not too hard to show this.

Now, you need to do the same for your collection of objects Sn . Notice, Sn is

the set {S1,S2,...} . Out of all the numerical values of Sn, can you think of the

least real number that is larger than all the Sn's? If you can figure out, a proof

should not be too far.
Sn

Although not the original intention of my question, I will go ahead and determine the Sup of this sequence. Since ##(S_n)## is monotonically "Fallend" non-increasing, ##S_1 = 1.5## is the sup.
 
  • #8
Well, but , under the standard notation, # \underset{n}{Sup} \ S_n # is:

Sup{S1,S2,..., Sn,...}
 
  • #9
Bacle2 said:
Well, but , under the standard notation, # \underset{n}{Sup} \ S_n # is:

Sup{S1,S2,..., Sn,...}

Thanks Bacle. I am now more familiar with this concept. I just wanted to make sure I am familiar with the notation I saw on Wikipedia and that I understood it correctly.
 

1. What is the definition of "Sup of Sequence: S_n"?

The Sup of Sequence, denoted as Sn, is the largest possible value that can be obtained by adding a finite number of terms from a given sequence. It represents the upper bound of the sequence.

2. How is Sup of Sequence calculated?

The Sup of Sequence is calculated by finding the maximum value among a set of finite terms in a sequence. This can be done by evaluating each term in the sequence and choosing the largest one.

3. Why is Sup of Sequence important in mathematics?

Sup of Sequence plays a crucial role in various mathematical concepts and applications. It helps in determining the convergence or divergence of a sequence, finding the limit of a sequence, and understanding the behavior of a sequence as n approaches infinity.

4. Can Sup of Sequence be infinite?

Yes, the Sup of Sequence can be infinite if the sequence itself is unbounded. This means that there is no upper limit to the values in the sequence, and therefore, the Sup of Sequence will also be infinite.

5. How is Sup of Sequence related to Inf of Sequence?

The Sup of Sequence and Inf of Sequence are two opposite concepts. While Sup of Sequence represents the upper bound, Inf of Sequence represents the lower bound of a sequence. Both of these values play a crucial role in understanding the behavior of a sequence.

Similar threads

  • Topology and Analysis
2
Replies
48
Views
3K
Replies
7
Views
1K
  • Calculus and Beyond Homework Help
Replies
17
Views
1K
  • Topology and Analysis
Replies
2
Views
1K
Replies
2
Views
1K
Replies
7
Views
2K
  • Set Theory, Logic, Probability, Statistics
Replies
2
Views
1K
  • Topology and Analysis
2
Replies
44
Views
5K
  • Calculus and Beyond Homework Help
Replies
7
Views
1K
Replies
1
Views
656
Back
Top