# Sequences convergent to 'cycles'

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).

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Basically, you have $L$ sequences "interweaved". I doubt there is any special name for them, but perhaps some analyst will know better than I. – Arturo Magidin Dec 17 '11 at 5:03
If I had naming rights, I would call it an $L$-colored sequence. – Bruno Joyal Dec 17 '11 at 5:11
I'd call it a limit-periodic sequence, but I'm not aware that a standard term exists for that. – alex.jordan Dec 17 '11 at 5:18
Thank you all for your comments. @alex.jordan, some research following your comment has lead me to believe that such sequences are called "asymptotically periodic". I will post an answer below. – matt Dec 18 '11 at 2:03

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: $$\lim_{n\to\infty}x_n - a_n = 0$$