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Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded?$ a_n = \frac{1}{2n + 3} $

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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

Baylor University

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.

01:50

Determine whether the sequ…

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plus determine whether this sequence is increasing, decreasing or perhaps not, Mon Atomic, then will determine whether it's founded. Now let's go ahead and show that the sequence is decreasing. So here, for any and bigger than or equal to one, we'd like to show that a N plus one is less than or equal to K N. Now this is equivalent to one over to and plus one plus three less than or equal to one over two M plus three. And this is true, since the denominator on the left hand side, it's smaller. Excuse me is larger. Larger denominator means smaller fraction. So by definition A M is decreasing. That's what it means to be decreasing. It means that the successor is no larger than the previous term. Now let's go ahead and show that is bounded. A N is bounded. Many ways to show this one of them, for example, is that we can say and is always bigger than zero. But it's always less than, for example, the first term, which is one over five. We know where over five is a one equals one over five, and we know that it's decreasing. So all the other ends, or less than one over five. On the other hand, we know that Anne is bigger than zero for any end, because positive over a positive is positive. That verifies this and these together imply that I am is a bounded sequence, so to summarize, it's founded and it's decreasing. That's our final answer.

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