In what sense is the sequence $(0,1,2,0,1,2,\ldots)$ convergent? In what sense is the sequence $(0,1,2,0,1,2,\ldots)$ convergent?
A problem I'm working on needs a concept of convergence which understands the fact that this sequence extended infinitely is convergent - at least in the sense that it is stable and orbits a fixed point.
The name may not be "convergence".
I can of course engineer something but I presume there is some accepted terminology I can use.
For example I want to describe that:
$$\lim_{n\to\infty}\left((1-2^{-n})+ (n\mod3)\right)$$ converges upon an orbit centred on the number $2$
 A: I would not use the word "convergent" in any sense!  This sequence is "periodic" with period 3.
A: You can take the already defined Cesàro summability, and sort of modify it to fit your needs. Let $(a_n)_{n=1}^\infty$ be your sequence. We can define a sequence to be Cesàro convergent if and only if
$$ \lim_{n\to \infty} \frac{1}{n}\sum_{i=1}^n a_i \in \mathbb R$$
That is, if the limit above exists, the sequence is Cesàro convergent. 
Notice, in your case, the limit above converges to $1$, since that is the average value of your sequence. So, in this sense, the sequence is "Cesàro convergent around 1".
A: The set of "accumulation points" (sometimes "cluster points") of the first sequence is $\{0,1,2\}$ and that set for the second sequence is $\{1,2,3\}$. This is because every neighborhood of one of the numbers, say $1$, has infinitely many sequence entries in that neighborhood.
This idea of the accumulation points is closely related to other ideas of "limit set" like attractors and limit cycles.
