Convergence Doubt: Is Most Repeated Value the Answer?

  • Context: Undergrad 
  • Thread starter Thread starter phoenix95
  • Start date Start date
  • Tags Tags
    Convergence Doubt
Click For Summary

Discussion Overview

The discussion revolves around the convergence of various sequences, particularly those composed of repeated values and alternating patterns. Participants explore the definitions and implications of convergence, limit superior, and limit inferior in relation to specific sequences.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants assert that a sequence consisting entirely of '1's converges to '1', while a sequence alternating between '1's and '0's diverges.
  • Questions are raised about the convergence of sequences with mixed patterns, such as '1, 1, 0, 1, 1, 0...' and '1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0...'.
  • One participant states that convergence depends on the long-term behavior of a sequence, indicating that initial terms do not affect convergence.
  • Another participant expresses confusion regarding the concepts of limit superior and limit inferior, seeking clarification.
  • A later reply suggests that the sequence with infinitely many '0's and '1's does not converge, referencing the definitions of limit superior and limit inferior.
  • There is a claim that a sequence with initial terms 'a' followed by 'b' converges to 'b', emphasizing the importance of long-term behavior.
  • Participants discuss the implications of having limsup and liminf values of '1' and '0', leading to conflicting interpretations about whether certain sequences converge or diverge.

Areas of Agreement / Disagreement

Participants express differing views on the convergence of specific sequences, particularly regarding the interpretation of limit superior and limit inferior. The discussion remains unresolved, with multiple competing perspectives on the convergence criteria.

Contextual Notes

Limitations include varying levels of understanding of convergence concepts among participants, and some participants express uncertainty about the definitions of limit superior and limit inferior.

phoenix95
Gold Member
Messages
81
Reaction score
23
Hello people,
In case if I am typing the question in the wrong forum please redirect me.
OK, here it goes... I have this stupid question:
Suppose we have a sequence 1, 1, 1, 1, 1, 1... It converges to '1'.
Consider 1, 0, 1, 0, 1, 0... it diverges right?
What about a sequence 1, 1, 0, 1, 1, 0... ?? Does it converge?
What about 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0,...?
What about any similar random sequence? does it converge to the most repeated value?

Thanks for any help in advance.
 
Physics news on Phys.org
Any time the limit superior of a sequence is not equal to the limit inferior, the limit will not exist.
 
  • Like
Likes   Reactions: phoenix95
phoenix95 said:
Hello people,
In case if I am typing the question in the wrong forum please redirect me.
OK, here it goes... I have this stupid question:
Suppose we have a sequence 1, 1, 1, 1, 1, 1... It converges to '1'.
Consider 1, 0, 1, 0, 1, 0... it diverges right?
What about a sequence 1, 1, 0, 1, 1, 0... ?? Does it converge?
What about 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0,...?
What about any similar random sequence? does it converge to the most repeated value?

Thanks for any help in advance.

Convergence by definition means getting close to a single value. So, no sequence with infinitely many 0's and 1's can converge.
 
  • Like
Likes   Reactions: phoenix95
I don't know what superior and inferior limits are, will you please explain?

How about this?
An infinite sequence with first few terms being 'a', and the rest all being 'b', like
a, a, a, a, a, b, b, b, b, b, b, b, b, b, b, b, b, b, b, b, b, b...... ??
Will it converge??
 
phoenix95 said:
I don't know what superior and inferior limits are, will you please explain?

How about this?
An infinite sequence with first few terms being 'a', and the rest all being 'b', like
a, a, a, a, a, b, b, b, b, b, b, b, b, b, b, b, b, b, b, b, b, b...... ??
Will it converge??

That sequence converges to b. Convergence depends on the long-term behaviour of a sequence. The initial terms do not affect this.
 
  • Like
Likes   Reactions: phoenix95
Well then, what are these superior and inferior limits?
 
phoenix95 said:
Well then, what are these superior and inferior limits?

You can google for this: limsup and liminf. But, unless you are studying real analysis seriously, I think these might be a bit difficult to digest.

The formal definition of convergence (again you can google this) is really the beginining of "rigorous" mathematics or "analysis". Most people find they have to work quite hard to get to grips with these concepts.

http://en.wikipedia.org/wiki/Limit_of_a_sequence#Formal_Definition
 
  • Like
Likes   Reactions: phoenix95
Ok, guys I got it... According to the problem, limsup=1 and liminf=0 isn't it? That's why the sequence converges...?? Well, thanks GFauxPas and PeroK...
 
phoenix95 said:
Ok, guys I got it... According to the problem, limsup=1 and liminf=0 isn't it? That's why the sequence converges...??
That's why the sequence doesn't converge (i.e., it diverges).
phoenix95 said:
Well, thanks GFauxPas and PeroK...
 
  • Like
Likes   Reactions: phoenix95

Similar threads

Undergrad Uniform convergence of family of functions with continuous index
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad Understanding the nth-term test for Divergence
  • · Replies 17 ·
Replies
17
Views
6K
Undergrad Infinite Series - Divergence of 1/n question
  • · Replies 5 ·
Replies
5
Views
3K
High School Understanding about Sequences and Series
  • · Replies 3 ·
Replies
3
Views
2K
High School How can hyperreal numbers make infinitesimals logically sound in calculus?
  • · Replies 24 ·
Replies
24
Views
7K
High School Proving the Existence and Value of a Derivative at a Point
  • · Replies 11 ·
Replies
11
Views
2K
Undergrad Convergence and divergence of series and sequences
  • · Replies 7 ·
Replies
7
Views
3K
Undergrad Question about uniform convergence in a proof
  • · Replies 1 ·
Replies
1
Views
3K
High School Difficulty with understanding whether 1/n converges or diverges
  • · Replies 11 ·
Replies
11
Views
6K
Undergrad Convergence of a recursively defined sequence
  • · Replies 18 ·
Replies
18
Views
4K