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Test the series for convergence or divergence.

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

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Diverges

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 7

Strategy for Testing Series

Sequences

Series

Campbell University

Harvey Mudd College

Idaho State University

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.

01:38

Test the series for conver…

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01:43

for this problem. We're going to use the Lim comparison test the and terms those air traditionally the terms that were starting with then the B in terms those the terms that were goingto be comparing our in terms, too, that being term should be somewhat similar to the in terms, in the sense that blame it as an approaches infinity of a n over bian. Hopefully that'LL be something that's non zero and finite. So usually the easiest way to accomplish that is just find the B in terms so that the this ratio that we mentioned is one that's what's happening here. One way to see that is that the AA minus one that we see in the A in terms of the plus one that we see in the end term those that can just be thrown out, so to speak as n goes to infinity as n goes to infinity, the n squared and in Cube they're going to totally dominate and the constants are going to be negligible in comparison. Okay, so that's that's the motivation for construction of this bien. And if you work it out then you'LL see that this limit is indeed one, okay. And what that means is that whatever happens, toa when we sum up, these be in terms. You'LL have the same behavior as if we sum up the's A in terms. So summing up, these be in terms. We're going to be getting the harmonic Siri's when we take in equals one to infinity of one over m the harmonic Siri's diverges, which means that we also must have that summing up the's A in Terms gives us divergence. And again, we note that this one here, the one is not the important part. The important part is that it's something that's non zero and finite that makes the limit comparison test applicable.

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