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Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded?$ a_n = \cos n $

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 1

Sequences

Series

Campbell University

University of Michigan - Ann Arbor

Idaho State University

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.

02:59

Determine whether the sequ…

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04:13

first, let's show that this sequence is not monotone IQ. So a one that's co sign one. And that's approximately point five four o three Now a two. This time you just do co sign of two. So let's go out and plug that in and then round off to a calculator. Negative point four one six one a three that's co signed three. And once again, back to the calculator negative point nine eight nine nine and then a four co sign. For now, let's go ahead and approximate that. Using the calculator once again, we're almost and that this will be enough for us to make to stop now. Originally, we could see that it was decreasing, and then it even decreased again, so it looks like it had a chance. But then it switched Teo increasing. Therefore, it cannot be decreasing because it increases. On the other hand, it can be increasing because it decreases. They're for a end is not monitor Nick. Now for the second question, we know that a N is less than or equal to one. An absolute value, says co sign is less than or equal to one and absolute value. So this sequence is bounded because we just showed that it's bounded by one. So it's not monotone yet. It's bounded. That's our final answer.

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