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(a) What is a sequence?(b) What does it mean to say that $ \lim_{n \to \infty} a_n = 8? $(c) What does it mean to say that $ \lim_{n \to \infty} a_n = \infty? $

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a. A sequence is an ordered list of numbers. It can also be defined as a function whose domain is the set of positive integers.b. The terms $a_{n}$ approach 8 as $n$ becomes large. In fact, we can make $a_{n}$ as close to 8 as we like by taking $n$ sufficiently large.c. The terms $a_{n}$ become large as $n$ becomes large. In fact, we can make $a_{n}$ as large as we like by taking $n$ sufficiently large.

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 1

Sequences

Series

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What is a convergent sequence? Give tqo examples

What is a convergent sequence? Give two examples

Baylor University

University of Michigan - Ann Arbor

University of Nottingham

Lectures

01:59

In mathematics, a series is, informally speaking, the sum of the terms of an infinite sequence. The sum of a finite sequence of real numbers is called a finite series. The sum of an infinite sequence of real numbers may or may not have a well-defined sum, and may or may not be equal to the limit of the sequence, if it exists. The study of the sums of infinite sequences is a major area in mathematics known as analysis.

02:28

In mathematics, a sequence is an enumerated 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 either the set of the natural numbers (for infinite sequences) or the set of the first "n" natural numbers (for a finite sequence). A sequence can be thought of as a list of elements with a particular order. Sequences are useful in a number of mathematical disciplines for studying functions, spaces, and other mathematical structures using the convergence properties of sequences. In particular, sequences are the basis for series, which are important in differential equations and analysis. Sequences are also of interest in their own right and can be studied as patterns or puzzles, such as in the study of prime numbers.

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a sequence. It is an order collection of numbers where were allowed tohave repeats of the same numbers. So ordered collection of numbers where repeats are allowed. What does it mean to say that limit as N goes to infinity of and is equal to eight? It means that as in gets bigger and gets closer and closer to eight. What does it mean to say that limiters and goes to infinity of an equals infinity? That means that as in gets larger and gets larger without found, you could make it as large as you want.

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