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Suppose you know that $ \left\{ a_n \right\} $ is a decreasing sequence and all its terms lie between the numbers 5 and 8. Explain why the sequence has a limit. What can you say about the value of the limit?

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$\left\{a_{n}\right\}$ is decreasing, so $5 \leq L<8$

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 1

Sequences

Series

Missouri State University

Harvey Mudd College

Boston College

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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71. Suppose you know that …

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Let $\left\{a_{n}\right\}$…

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Limits of Sequences If the…

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for this problem. You're given that hey and is decreasing, and here you see that it's between five and eight for each end. This means that the sequence is bounded. You have an upper and lower bound. That means it's bounded. Now, why must this sequence have a limit? Let's answer that using a the room since A M is bounded and monotone by the monotone sequence, their own and converges. On the other hand, we see that the sequence is bounded between five and eight, so it's not possible for the limit to go outside of the bounds. So we can say that the limit is in between five and eight does not necessarily have to be fiber eight again five and eight or bounds. It's not possible for the sequence escaped the bounds. Therefore, the limit must stay inside that lower upper bounds. So the limits between five and eight that's our final answer

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