Week 1
... Proof. (i) says that N is not bounded above. Assume to the contrary that it is. Then α = sup N will exist. Since α − 1 is not an upper bound of N, there will be n ∈ N : α − 1 < n. Then α < n + 1. Since n + 1 ∈ N this contradicts the fact that α is an upper bound. (ii) follows from (i) by letting x = ...
... Proof. (i) says that N is not bounded above. Assume to the contrary that it is. Then α = sup N will exist. Since α − 1 is not an upper bound of N, there will be n ∈ N : α − 1 < n. Then α < n + 1. Since n + 1 ∈ N this contradicts the fact that α is an upper bound. (ii) follows from (i) by letting x = ...
Recursive Rules for Sequences
... So far in this chapter you have worked with explicit rules for the nth term of a sequence, such as an = 3n º 2 and an = 3(2)n. An explicit rule gives an as a function of the term’s position number n in the sequence. In this lesson you will learn another way to define a sequence—by a recursive rule. ...
... So far in this chapter you have worked with explicit rules for the nth term of a sequence, such as an = 3n º 2 and an = 3(2)n. An explicit rule gives an as a function of the term’s position number n in the sequence. In this lesson you will learn another way to define a sequence—by a recursive rule. ...
Random Number Generation
... Neumann and Metropolis 1940) ◦ Start with a four digit positive integer Z0. ◦ Square Z0 to get an integer with up to eight digits (append zeros if less than eight). ◦ Take the middle four digits as the next four digit integer Z1. ◦ Place a decimal point to the left of Z1 to form the first “U(0,1)” o ...
... Neumann and Metropolis 1940) ◦ Start with a four digit positive integer Z0. ◦ Square Z0 to get an integer with up to eight digits (append zeros if less than eight). ◦ Take the middle four digits as the next four digit integer Z1. ◦ Place a decimal point to the left of Z1 to form the first “U(0,1)” o ...
12-1 Notes 9/6/16 (Introduction to Sequences)
... terms of the sequence as the range. Values in the domain are called term numbers and are represented by n. Instead of function notation, such as a(n), sequence values are written by using subscripts. The first term is a1, the second term is a2, and the nth term is an. Because a sequence is a functio ...
... terms of the sequence as the range. Values in the domain are called term numbers and are represented by n. Instead of function notation, such as a(n), sequence values are written by using subscripts. The first term is a1, the second term is a2, and the nth term is an. Because a sequence is a functio ...
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.