§1. Basic definitions Let IR be the set of all real numbers, while IR
... [an , bn ]. Assume the opposite: ∃η ∈ [an , bn ] ∀n and η 6= ξ. Then by Theorem 3.6 bn − an ≥ |ξ − η| > 0 ∀n ⇒ ...
... [an , bn ]. Assume the opposite: ∃η ∈ [an , bn ] ∀n and η 6= ξ. Then by Theorem 3.6 bn − an ≥ |ξ − η| > 0 ∀n ⇒ ...
ON THE STRONG LAW OF LARGE NUMBERS FOR SEQUENCES
... for all choices of 1 ~ n < m ~ N, for all constants {af, ... , am}, and for all permutations TC of the integers {1, ... , m}. A sequence of random elements {~, n > 1} is said to be p-orthogonal (1 ~ p < (0) if {V1, ... , VN} is p-orthogonal for all N ;?: 2. The notion of p-orthogonality was introduc ...
... for all choices of 1 ~ n < m ~ N, for all constants {af, ... , am}, and for all permutations TC of the integers {1, ... , m}. A sequence of random elements {~, n > 1} is said to be p-orthogonal (1 ~ p < (0) if {V1, ... , VN} is p-orthogonal for all N ;?: 2. The notion of p-orthogonality was introduc ...
Asymptotic Equality and Inequality
... described next. A perhaps more intuitive definition that works in the important case of sequences of positive numbers will be given in Exercise *** below. The definition will be followed by a series of exercises each of which can be solved in a few lines given the exercises preceding it; these exerc ...
... described next. A perhaps more intuitive definition that works in the important case of sequences of positive numbers will be given in Exercise *** below. The definition will be followed by a series of exercises each of which can be solved in a few lines given the exercises preceding it; these exerc ...
LESSON PLAN FOR THE TEACHER
... Each row is symmetrical, beginning with 0, then the diagonals go consecutive integers, even integers, then triangular numbers, etc. (Discuss patterns seen by students) Reflection on Learning: Summarize the possible methods that you can use to determine an explicit formula for a sequence. Read respo ...
... Each row is symmetrical, beginning with 0, then the diagonals go consecutive integers, even integers, then triangular numbers, etc. (Discuss patterns seen by students) Reflection on Learning: Summarize the possible methods that you can use to determine an explicit formula for a sequence. Read respo ...
Series and Sequences - Answer Explanations
... term to produce the next term. The number that is added to each term to produce the next term is known as the common difference. If you subtract the previous term from any term in the series, the result is equal to the common difference. For instance, 5, 7, 9, 11, 13, 15… is an arithmetic series wit ...
... term to produce the next term. The number that is added to each term to produce the next term is known as the common difference. If you subtract the previous term from any term in the series, the result is equal to the common difference. For instance, 5, 7, 9, 11, 13, 15… is an arithmetic series wit ...
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.