# Convergence doubt

1. Oct 22, 2014

### 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.

2. Oct 22, 2014

### GFauxPas

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

3. Oct 22, 2014

### PeroK

Convergence by definition means getting close to a single value. So, no sequence with infinitely many 0's and 1's can converge.

4. Oct 22, 2014

### 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??

5. Oct 22, 2014

### PeroK

That sequence converges to b. Convergence depends on the long-term behaviour of a sequence. The initial terms do not affect this.

6. Oct 22, 2014

### phoenix95

Well then, what are these superior and inferior limits?

7. Oct 22, 2014

### PeroK

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

8. Oct 22, 2014

### 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....

9. Oct 22, 2014

### Staff: Mentor

That's why the sequence doesn't converge (i.e., it diverges).

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