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Use any test to determine whether the series is absolutely convergent, conditionally convergent, or divergent.

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

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 6

Absolute Convergence and the Ratio and Root Tests

Sequences

Series

Harvey Mudd College

Baylor University

University of Nottingham

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.

06:52

Use any test to determine …

05:08

02:26

02:42

00:49

Determine whether the seri…

02:45

02:27

Let's show that this series is absolutely convergence, so here will go ahead and look at a similar Siri's. But this time will take absolute value. So here there's absolute value. So all that will do for us is just get rid of the negative one to the end power. And let me let me explain why here? This is because and square in our town of n are both positive in here. So technically, Arc Tan could be negative. You think of the graph of this, but as long as and is positive than Arc Tan is positive, so that justifies Why weaken, drop the absolute values and then this Siri's. We know this is less than or equal to the largest that the Ark Tan could be. Is this horizontal Assam towed at pi over too. So just go ahead and replace Arc tan with pie over too. And here you can pull out, pull off their constant. So we're using a few test here. You could see from the inequality that were setting ourselves up to use the comparison test. So now we have to look at our upper bound this new series over here blue and that's a P Siri's. So the Siri's converges you would apply the Pee test with P equals two, and that is bigger than one. So the Siri's will converge. So now the upper bound converges. So by the comparison test, the lower bound also converges, and we conclude that the Siri's that the original Siri's is absolutely commercial. So let me just go ahead and call this instead of writing this whole thing down again. Let's just call this star. The Serie Star converges. Excuse me, is absolutely commercial.

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