Convergence Doubt: Is Most Repeated Value the Answer?

[phoenix95]

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.

 

[GFauxPas]

Any time the limit superior of a sequence is not equal to the limit inferior, the limit will not exist.

 

[PeroK]

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.

 

[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??

 

[PeroK]

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.

 

[phoenix95]

Well then, what are these superior and inferior limits?

 

[PeroK]

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

 

[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...

 

[Mark44]

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).

Well, thanks GFauxPas and PeroK...