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Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded?$ a_n = 2 + \frac{(-1)^n}{n} $

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 1

Sequences

Series

Campbell University

Oregon State University

Harvey Mudd College

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.

03:59

Determine whether the sequ…

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02:59

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02:25

Let's consider the sequence given by N now. The first part is determine whether this thing is increasing, decreasing. Or perhaps it's not even Mon Atomic. Well, let's just write out a few terms here. A one that's just two minus one equals one. Now, how about a two? This one will be to plus a half. So five halfs. So from the first term, to the seconds her we saw increase, how about from the second terms of the third term? This time when and his three will have to minus the thirds, This is five thirds, and this is a decrease. So because of the term from a one to a two, this showed us that it was not decreasing because it increase. And from a two two a three, we see that it's not increasing. Therefore, the sequence A M is neither increasing nor decreasing, so it's not monitor tonic. Now let's go to the next part. We see that we start off at two, and then we're adding a term that goes to zero. So for this problem, it might be best to just noticed that the limit of an equals two so a N is bounded, and this is just coming from the fact that convergent sequences are bounded. This is an important that and we're using that to show and his bounded because it converges. And that's our final answer, not mon atomic, but it is bounding.

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