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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.
A sequence could be a finite sequence from a data source or an infinite sequence from a discrete dynamical system. Such a discrete function could be defined explicitly by a list (if its domain is finite), or by a formula for its general term, or it could be given implicitly by a recurrence relation or difference equation.
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 .
Convolution. Cauchy product –is the discrete convolution of two sequences; Farey sequence – the sequence of completely reduced fractions between 0 and 1; Oscillation – is the behaviour of a sequence of real numbers or a real-valued function, which does not converge, but also does not diverge to +∞ or −∞; and is also a quantitative measure for that.
A possible solution to this problem is that, instead of defining a numeric utility for each infinite outcome sequence, we just define the preference relation between two infinite sequences. We say that agent i {\displaystyle i} (strictly) prefers the sequence of outcomes y t {\displaystyle y_{t}} over the sequence x t {\displaystyle x_{t}} , if ...
If I and J are finite sets, then the presentation is called a finite presentation; a module is called finitely presented if it admits a finite presentation. Since f is a module homomorphism between free modules , it can be visualized as an (infinite) matrix with entries in R and M as its cokernel .
A renewal system is defined to be the set of all infinite concatenations of some fixed finite collection of finite words. Subshifts of finite type are identical to free (non-interacting) one-dimensional Potts models ( n -letter generalizations of Ising models ), with certain nearest-neighbor configurations excluded.
If both are finite it is said to be a finite presentation. A group is finitely generated (respectively finitely related, finitely presented) if it has a presentation that is finitely generated (respectively finitely related, a finite presentation). A group which has a finite presentation with a single relation is called a one-relator group.