Difference equations - Cambridge University Press
... However, if the sequence represented by the difference equation is arithmetic or geometric, we can solve the difference equation using our knowledge of arithmetic or geometric sequences. Thus we can save ourselves much work if we can recognise, right from the start, whether the sequence represented ...
... However, if the sequence represented by the difference equation is arithmetic or geometric, we can solve the difference equation using our knowledge of arithmetic or geometric sequences. Thus we can save ourselves much work if we can recognise, right from the start, whether the sequence represented ...
SEQUENCES, CONTINUED Definition 3.13. A sequence {sn} of real
... Recall that S R is the set of extended real number subsequential limits of fsn g. Theorem 3.17. (a) s 2 S and s 2 S. (b) For each x > s there exists N 2 N such that sn < x for n N . For each x < s there exists N 2 N such that sn > x for n N . Moreover, s and s are the only numbers with properties (a ...
... Recall that S R is the set of extended real number subsequential limits of fsn g. Theorem 3.17. (a) s 2 S and s 2 S. (b) For each x > s there exists N 2 N such that sn < x for n N . For each x < s there exists N 2 N such that sn > x for n N . Moreover, s and s are the only numbers with properties (a ...
Metric Spaces
... functions of one or more real variables, that is functions defined on a subset of the real line, or the plane, or ordinary 3-space or, more generally, n-dimensional Euclidean space. The real line, plane, etc. are special cases of the general concept of “metric space’’ which is introduced in this cha ...
... functions of one or more real variables, that is functions defined on a subset of the real line, or the plane, or ordinary 3-space or, more generally, n-dimensional Euclidean space. The real line, plane, etc. are special cases of the general concept of “metric space’’ which is introduced in this cha ...
Sequence
In mathematics, a sequence is an ordered collection of objects in which repetitions are allowed. Like a set, it contains members (also called elements, or terms). The number of elements (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Formally, a sequence can be defined as a function whose domain is a countable totally ordered set, such as the natural numbers.For example, (M, A, R, Y) is a sequence of letters with the letter 'M' first and 'Y' last. This sequence differs from (A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be finite, as in these examples, or infinite, such as the sequence of all even positive integers (2, 4, 6,...). In computing and computer science, finite sequences are sometimes called strings, words or lists, the different names commonly corresponding to different ways to represent them into computer memory; infinite sequences are also called streams. The empty sequence ( ) is included in most notions of sequence, but may be excluded depending on the context.