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Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded?$ a_n = n(-1)^n $

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the sequence is not bounded.

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 1

Sequences

Series

Harvey Mudd College

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.

02:25

Determine whether the sequ…

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for this sequence will actually show that it's not monotone and that it's also not bounded now. The way to show that it's not monotone is to show that it's not increasing and decreasing, so a one equals one. Excuse me, a negative one. A two is equal to two, so that's an increase. However, a three equals minus three, and that's a decrease. So the sequence is not decreasing because at some point there's the increase. On the other hand, the sequence is not increasing because there's a decrease. So therefore, a N is neither increasing or decreasing, so it's not monotone now. On the other hand, let's show that it's not bounded. So instead of looking at all the ends, let's just look at the ends of the form to. And so, for example, I mean, like a two, a four, a six and so on notice that these are all just equal to two floor six and so on, and therefore the limit of a to end is just equal to the limit of to end. These air limits is and goes to infinity and the limit of two, and it's just infinity, so there's no way that the sequence could be bounded because we have this. Some of the terms are getting larger and larger, closer to infinity. So a N is not bounded, and that's our final answer, not bounded in that monotone.

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