TITLE: Sequences convergent to 'cycles'
QUESTION [5 upvotes]: Consider sequences $(x_n)_{n=1}^\infty\subset\mathbb R$. Is there a name for the following property?

There exists $L\in\mathbb N$ such that:
$$\lim\limits_{k\rightarrow\infty}x_{(kL+m)}=x^\ast_m$$
for $m\in\{0,1,2,\dots,(L-1)\}$.

Here the $x^\ast_m$'s are not necessarily equal.
As an example, the sequence $x_n=(-1)^n +\frac n{n+1}$ has $x_{2n}\rightarrow 2$ and $x_{2n+1}\rightarrow 0$ as $n\rightarrow\infty$. In this case $L=2$ (choosing $L$ to be minimal).

REPLY [4 votes]: A sequence with the described property is called an asymptotically periodic sequence. 

A sequence is asymptotically periodic if its terms approach those of a periodic sequence. That is, the sequence $x_1,x_2,x_3,\dots$ is asymptotically periodic if there exists a periodic sequence $a_1,a_2,a_3,\dots$ for which:
  $$\lim_{n\to\infty}x_n - a_n = 0$$

Source: http://en.wikipedia.org/wiki/Periodic_sequence#Generalizations