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Test the series for convergence or divergence.

$ \displaystyle \sum_{n = 1}^{\infty} ( - 1)^n \frac {n^2 - 1}{n^3 + 1} $

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 7

Strategy for Testing Series

Sequences

Series

Baylor University

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.

01:52

Test the series for conver…

01:22

01:38

03:37

01:16

05:03

There's some that we're looking at is alternating sign. So we'd be thinking about the alternating signed test the alternating signed test. You take the absolute value of your terms Case on this case, we're going to get this guy. So then, as long as, Ah, this is eventually decreasing to zero. Then, since we're alternating signs, does this minus one? Then then that means that this sum, this whole thing here, is going to converge. So think of this is a rational function with the the power of the denominator being greater than the power of the numerator. We know that eventually this is goingto be decreasing. Okay? And as we mentioned, since the power, the denominator is greater than the power of the numerator. We know that it'LL approach zero, which is what we want. So this goes to zero as and goes to infinity and as we also commented on property of all rational functions, is that they will eventually be monotone. So the alternating signed test is applicable here and absolute value of our terms in consideration. New approach zero. So we get convergence in this case

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