You may have seen a classification of sequences into:
(c) periodic / oscillating
We start using more ... (dot dot dot) with sums of terms, investigating whether we're approaching a limit or not -- the on-ramp to calculus.
High school texts often skimp on (d) chaotic because that's a new branch since the 1800s, and as we all know, anything 1900s is quasi-verbotten in that locked down slave ship of a sad sorry discipline called K-12 math teaching -- nothing much to do with actual math.
In Gnu Math, we skip from the 1800s to around now, which is a big shock to some teachers, but a real boost and eye-opener to those students lucky enough to "get it".
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