Convergence Doubt: Is Most Repeated Value the Answer?

In summary, the conversation discusses the concept of convergence in sequences and how it depends on the long-term behavior of the sequence. The superior and inferior limits are defined and their role in determining convergence is explained. The example of a sequence with alternating values and its convergence is also discussed.
  • #1
phoenix95
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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.
 
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  • #2
Any time the limit superior of a sequence is not equal to the limit inferior, the limit will not exist.
 
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  • #3
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.
 
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  • #4
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??
 
  • #5
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.
 
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  • #6
Well then, what are these superior and inferior limits?
 
  • #7
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
 
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  • #8
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...
 
  • #9
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...
 
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FAQ: Convergence Doubt: Is Most Repeated Value the Answer?

1. What is convergence doubt?

Convergence doubt is a term used in statistics and data analysis to describe uncertainty or skepticism about the results of a study or experiment when the data appears to be converging on a single value.

2. What is the most repeated value in convergence doubt?

The most repeated value in convergence doubt refers to the value that appears most frequently in the data set. This value may also be referred to as the mode.

3. Is the most repeated value always the most accurate answer?

No, the most repeated value is not always the most accurate answer. Convergence doubt suggests that the data may be converging on a single value, but it does not guarantee its accuracy. Other factors, such as sample size and outliers, should also be considered when determining the most accurate answer.

4. How can convergence doubt be addressed?

Convergence doubt can be addressed by conducting further analysis and testing to validate the results. It is also important to carefully examine the data for any outliers or anomalies that may be skewing the results.

5. Can convergence doubt be avoided?

Convergence doubt is a natural part of the scientific process and cannot always be avoided. However, careful experimental design and data collection can help minimize convergence doubt and increase the validity and reliability of the results.

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