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Determine whether the sequence converges or diverges. If it converges, find the limit.$ \left \{ \frac {(2n - 1)!}{(2n + 1)!}\right \}$

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 1

Sequences

Series

Missouri State University

Oregon 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.

00:55

Determine whether the sequ…

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05:03

we're the same thing that we're doing before. It's just written a little bit differently now to the A end. Terms that were looking at her two in minus one fact will divided by two in plus one fact oil and wondered the same thing we were doing before. We want to look at the limit as in goes to infinity of a n well, whether or not that women exists. And if that women exists and is finite than our sequences, said that that is said to converge. Excuse me. Otherwise, that is said to diverge. So before we take the limit, weaken rewrite this expression a little bit. So this is two in minus one. Fact will and then another way that we can write too. In plus one, factorial is two in minus one factorial times two in times to end plus one. And this is a convenient way to write it because now we have ah, nice cancellation. That happens right here. And we see that this simplifies to one over to in times two and plus one. And now when we do limit as n goes to infinity of a n r. Expression becomes easy to work with. We just have one over to end times two and plus one. So certainly the denominator is going to go to infinity. The numerator is just one. So this is one over infinity, which is zero. So our sequence does converge, converges to zero.

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