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Here, is the set of integers, and is the set of non-negative integers. Here, all letters represent integers. We say that a map : is piecewise linear if can be defined by various affine functions with integer coefficients on each of finitely many pieces, where each piece is defined by a finite set of linear inequalities with integer coefficients.
An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.
The infinite sequence of additions expressed by a series cannot be explicitly performed in sequence in a finite amount of time. However, if the terms and their finite sums belong to a set that has limits , it may be possible to assign a value to a series, called the sum of the series .
A sequence obeying the order-d equation also obeys all higher order equations. These identities may be proved in a number of ways, including via the theory of finite differences. [9] Any sequence of + integer, real, or complex values can be used as initial conditions for a constant-recursive sequence of order +.
Equivalently, a sequence is a harmonic progression when each term is the harmonic mean of the neighboring terms. As a third equivalent characterization, it is an infinite sequence of the form 1 a , 1 a + d , 1 a + 2 d , 1 a + 3 d , ⋯ , {\displaystyle {\frac {1}{a}},\ {\frac {1}{a+d}},\ {\frac {1}{a+2d}},\ {\frac {1}{a+3d}},\cdots ,}
A finite game is played for the purpose of winning, an infinite game for the purpose of continuing the play. Finite games are those instrumental activities - from sports to politics to wars - in which the participants obey rules, recognize boundaries and announce winners and losers.
A (one-sided) shift of finite type is the set of all sequences, infinite on one end only, that can be made up of the letters ,,, like ,, ….A (two-sided) shift of finite type is similar, but consists of sequences that are infinite on both ends.
The numerical value of an infinite continued fraction is irrational; it is defined from its infinite sequence of integers as the limit of a sequence of values for finite continued fractions. Each finite continued fraction of the sequence is obtained by using a finite prefix of the infinite continued fraction's defining sequence of integers.