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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.

Let $(b_n)$ be positive and increase to infinity. If the series ${\sum }_{n\ge 1}{x}_n/b_n$ converges, thenThis follows from summation by parts: write $x_k/b_k$ as a convergent series and average its tails with the increasing weights $b_k$.

A lattice sum is a series indexed by the points of a lattice. For example, comparison with annular lattice-point counts shows that ${\sum }_{(m,n)\ne (0,0)}(m^2+n^2{)}^{-s}$ converges when $s>1$.

For terms $(a_n)$, the $n$th partial sum is $S_n={\sum }_{j=1}^n{a}_j$. A series converges when its sequence of partial sums converges.