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In most cases, the k-kernel of a sequence is infinite. However, if the k-kernel is finite, then the sequence s(n) is k-automatic, and the converse is also true. This is due to Eilenberg. [8] [9] [10] It follows that a k-automatic sequence is necessarily a sequence on a finite alphabet.
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 .
The Cauchy product may apply to infinite series [1] [2] or power series. [3] [4] When people apply it to finite sequences [5] or finite series, that can be seen merely as a particular case of a product of series with a finite number of non-zero coefficients (see discrete convolution). Convergence issues are discussed in the next section.
This definition covers several different uses of the word "sequence", including one-sided infinite sequences, bi-infinite sequences, and finite sequences (see below for definitions of these kinds of sequences). However, many authors use a narrower definition by requiring the domain of a sequence to be the set of natural numbers. This narrower ...
The basic open sets of the product topology are cylinder sets.These can be characterized as: If any finite set of natural number coordinates I={i} is selected, and for each i a particular natural number value v i is selected, then the set of all infinite sequences of natural numbers that have value v i at position i is a basic open set.
For an infinite sequence, one is often more interested in the long-term behaviors of the sequence than the behaviors it exhibits early on. In which case, one way to formally capture this concept is to say that the sequence possesses a certain property eventually, or equivalently, that the property is satisfied by one of its subsequences (), for some .
A (necessarily finite) sequence (,, …,) of elements of that contains no pair with < is usually called a bad sequence. The tree of bad sequences T X {\displaystyle T_{X}} is the tree that contains a vertex for each bad sequence, and an edge joining each nonempty bad sequence ( x 1 , … , x n − 1 , x n ) {\displaystyle (x_{1},\ldots ,x_{n-1 ...
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