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A power series encoding a sequence through its coefficients. It turns combinatorial splitting and concatenation rules into algebraic identities. A counting generating function is not necessarily a probability generating function: its coefficients need not sum to one.

A geometric progression is a sequence whose successive terms have a constant ratio: $a_n=a{r}^n$ for constants $a$ and $r$.

Every Goodstein sequence reaches zero. Before zero, the ordinal rank of a Goodstein term of each step is strictly smaller than the preceding one. An infinite sequence would contradict well-foundedness of the ordinals below epsilon zero.

An exhaustion of a countable graph is an increasing sequence of finite connected graph vertex sets whose union is the whole graph vertex set. In a locally finite graph, every prescribed finite graph path together with all graph neighbours of its visited graph vertices lies in a sufficiently large exhaustion set. This permits coupling finite-volume and infinite-volume simple random walks up to a finite stopping time.

Finite rooted trees are well-quasi-ordered by rooted-tree homeomorphic embedding. Consequently every infinite sequence contains an earlier tree embedding into a later one. Size-controlled finite forms follow by applying König infinity lemma to the finitely branching tree of bad prefixes.

On a two-sided full shift, the left shift is the homeomorphismIts inverse sends $z(j)$ to $z(j-1)$. On a one-sided sequence space, the same forward shift is generally not invertible.

A real sequence $(x_n)$ has limit of a sequence $L\in \mathbb{R}$ when for every $\epsilon >0$ there is $N$ such that $|x_n-L|<\epsilon$ for all $n\ge N$. Its limit of a sequence is unique. Extended-real convergence to $-\infty$ means that every real upper bound eventually exceeds all the terms; convergence to $+\infty$ is the corresponding lower-bound condition. The extended-real Fekete lemma is an example where the limit of a sequence need not be finite.

A point is minimal when its forward orbit closure is a minimal dynamical system. This does not require the point to be a fixed point. In a finite-alphabet full shift, minimal points are exactly the uniformly recurrent sequences.