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Determine whether the series converges or diverges.$ \displaystyle \sum_{k = 1}^{\infty} \frac {(2k - 1)(k^2 - 1)}{(k + 1)(k^2 + 4)^2} $

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 4

The Comparison Tests

Sequences

Series

Missouri State University

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.

05:59

Determine whether the seri…

00:47

01:22

00:21

Determine whether each ser…

00:30

01:28

01:39

01:08

00:54

Let's use the Lim comparison test. Let's call this term here a k and then let's call B k T B two. Okay, Cube And then here, Kate in the fifth. So I just get these powers by looking at the largest power in the numerator a three. And then here I have four, and then one more. That's a five. And then I could simplify this if I wanted to. And we know that there's some converges. This is just a P Siri's with P equals two. So now let's use limit comparison. So we look at the limit. Kay goes to infinity, a k a. Over b. K. So that's here's r a K and then we'LL go ahead and multiply this bye bye flipping B and then here let's go ahead and multiply this out. So, of course, here we could also cancel three of those cases as we did before. I should have used this. So in the numerator, Scott and more supply this all outlets foil this. That's two K and then we have to the fifth and then here, minus king of the fourth and then minus two cake cute plus case. Where in the denominator king of the fourth to face Where? Plus one And then let's just go ahead to the side and simplify this denominator that'LL be killing us is two k cube plus que plus Kato The fourth two case Where plus one So that's in our denominator. And then here we're almost done after we've right this denominator, Let's go ahead and divide top and bottom. Bye, kid of the fifth. So here, let's see Divide by Kato the fifth. So this is once on the numerator and also on the denominator. So divide this by King of the Fifth and I'LL write this all the way over here We'LL have limit two minus one over Kay minus two over. Case Weird plus one over K Cube. That's our numerator and the denominator one plus one over. Okay, plus two over Kate Square plus two over K cute plus one over Kato, the fourth plus one over Kate in the fifth. Now, as we take the limit, all of these fractions with the K and the denominator go to zero and we're just left over with two divided by one. So at this point, we can go ahead and used what your book calls the limit comparison test. So we've computed the limit and we got a number that's between zero and infinity, so we can go ahead and use this test by the limit comparison. This Siri's one in our question. That should be okay, not a end. And here, and that's our final answer.

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