Sup of a sequence


by Bachelier
Tags: sequence
Bachelier
Bachelier is offline
#1
Feb27-13, 10:12 PM
P: 376
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
Phys.Org News Partner Science news on Phys.org
Review: With Galaxy S5, Samsung proves less can be more
Making graphene in your kitchen
Study casts doubt on climate benefit of biofuels from corn residue
rasmhop
rasmhop is offline
#2
Feb28-13, 05:56 AM
P: 418
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.
Bacle2
Bacle2 is offline
#3
Mar1-13, 07:41 PM
Sci Advisor
P: 1,168
No Sup for you! NEXT.

Bacle2
Bacle2 is offline
#4
Mar1-13, 08:52 PM
Sci Advisor
P: 1,168

Sup of a sequence


Quote Quote by Bachelier View Post
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
Bachelier
Bachelier is offline
#5
Mar1-13, 09:16 PM
P: 376
Quote Quote by rasmhop View Post
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) :)




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_s...limit_inferior

for instance see the image below:


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.
Bachelier
Bachelier is offline
#6
Mar1-13, 09:23 PM
P: 376
Quote Quote by Bacle2 View Post
No Sup for you! NEXT.
Good one. I prefer Salads anyway.
Bachelier
Bachelier is offline
#7
Mar1-13, 09:41 PM
P: 376
Quote Quote by Bacle2 View Post
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.
Bacle2
Bacle2 is offline
#8
Mar4-13, 04:13 PM
Sci Advisor
P: 1,168
Well, but , under the standard notation, # \underset{n}{Sup} \ S_n # is:

Sup{S1,S2,......, Sn,......}
Bachelier
Bachelier is offline
#9
Mar4-13, 11:43 PM
P: 376
Quote Quote by Bacle2 View Post
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.


Register to reply

Related Discussions
Determining if a sequence is convergent and/or a Cauchy sequence Calculus & Beyond Homework 3
Determine output sequence given input sequence and state table Engineering, Comp Sci, & Technology Homework 6
!Showing a sequence is less than another sequence (sequences and series question) Calculus & Beyond Homework 1
Sequence analysis of the Fibonacci sequence using matrices? Calculus & Beyond Homework 7
energy of a sequence - absolutely summable sequence or square summable sequence Engineering, Comp Sci, & Technology Homework 0