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Determine whether the series is absolutely convergent or conditionally convergent.

$ \displaystyle \sum_{n = 1}^{\infty} \frac {( - 1)^n}{n^3 + 1} $

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Absolutely Convergent

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 6

Absolute Convergence and the Ratio and Root Tests

Sequences

Series

Oregon State University

Harvey Mudd College

Baylor 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.

00:56

Determine whether the seri…

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03:27

let's go ahead and show that the series will be absolute conversion. Sure. So here we should look at this. Siri's obtained by taking the absolute value. Now the sequels, however, this is less than or equals who one over and tube. Why is this true? Therefore, that's true. And the Siri's convergence bye pee test with P equals three, which is bigger than one. That's the Peter the exponents. So by the comparison test absolute value here, the Siri's of absolute values conversions now. Therefore, we've just shown that the Siri's is absolutely commercial, and that's our finalists.

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