Does an alternating sequence converge or diverge or none?

Question

How come this sequence does not approach any limit?

$\{\max((-1)^n,0)\}^\infty _{n=1} : {0,1,0,1,0,1,0,1,...}$

I read that since this alternates between 0 and 1 this does not approach any limit. Hence not convergence. Is it safe to say that it does not diverge either?

Since:

A sequence can be divergent by having terms that increase (decrease) without limit. Example:
2,4,8,16,32,64,...

Does all sequences that alternates not approach a limit? For example this one too:

$3,1,3,1,3,1,3,1...$

So what name does these sequences have and what does it mean? It's neither convergent nor divergent. How do you prove that it is neither?

For example in the Collatz problem you "always" run into cycles (sub-sequences) that are similar to this.

Answer 1

Every sequence of real numbers either converges or diverges. This is trivial, since divergence means the opposite of convergence.

And the sequences that you mentioned diverge. That is, there is no real number $L$ such that$$(\forall\varepsilon>0)(\exists p\in\mathbb{N})(\forall n\in\mathbb{N}):n\geqslant p\implies|L-a_n|<\varepsilon.$$

You are wrong when when assert that that's what happens in the Collatz problem. If we knew that, it wouldn't be a problem anymore.

Answer 2

You have missed the definition of a divergent sequence.

A divergent sequence does not have to be unbounded, it simply does not have a limit.

$$ 1,0,1,0,1,0,... $$ does not converge so it is divergent.

Simply put, if a sequence is not convergent we call it divergent regardless of its other properties.