Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Use the Ratio Test to determine whether the series is convergent or divergent.

$ \displaystyle \sum_{n = 1}^{\infty} \frac {( - 2)^n}{n^2} $

Get the answer to your homework problem.

Try Numerade free for 7 days

Like

Report

Diverges

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 6

Absolute Convergence and the Ratio and Root Tests

Sequences

Series

Missouri State University

Harvey Mudd College

University of Michigan - Ann Arbor

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.

03:44

Use the Ratio Test to dete…

03:31

03:41

0:00

02:56

03:10

let's use the ratio test to determine whether this series converges or diverges. Now, the ratio test involves looking at this term here a n So in our case, this is negative two to the end over and squared. So we're interested in the answer to this limit. Here we look at absolute value a n plus one over a n and notice that the plus one, the one is being added to the n, not a n. So let's just go ahead and evaluate numerator and denominator here, so and plus one maybe do this in red, right? This is negative. Two and plus one right over n plus one square. Yeah, and then in blue and all right, there it is down there just using our formula up here for a M. Uh huh. And then now it's maybe clean this up a little bit by writing it as a product, and I could lose the absolute value. And when I do that, I'll get remember these negative signs here. So and I also I'll rewrite this as two times two to the end. Yeah, Now we see by doing this the reason for doing this trick up here is so that we could cancel the to to the ends. And now this is all equal to the limit and goes to infinity two and squared over and plus one square. This limit is of the form infinity over infinity. But you could do some algebra to simplify this. You could even use low petals. Role here have I used low petals rule. So you differentiate top and bottom with respect to end. So the numerator becomes foreign, the denominator becomes too and plus one. And here you'd still have infinity over infinity so you could use Low Patel once more low petals rule. So the numerator becomes just before the denominator becomes the two. That limit is two. However, that's bigger than one. So going on to the next page, we would be able to conclude that this series here is going to diverge. So let's let me write that out on the last page since the limit, How yeah, is bigger than one. The series okay, diverges by the ratio test, and that's our final answer.

View More Answers From This Book

Find Another Textbook

01:44

'answer the question properly '

01:31

'Find Reciprocal of please help me-5 3 6 4X5 +15 1x5 +3 8'

01:09

'answer the question please'

01:53

'find the measure of unknown angles x, y and z '

01:45

"Simplify the given equation:23 3[25 + | '3 (64) + (27) X …

01:21

'please do this question in your notebook'