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Test the series for convergence or divergence.

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

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Converges

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 7

Strategy for Testing Series

Sequences

Series

Oregon State University

Harvey Mudd College

Baylor University

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.

01:17

Test the series for conver…

01:52

01:22

01:43

01:38

03:59

There's another limit comparison problem that's going to be the A in terms that we're using and be in terms We want to be thinking about what happens when n goes to infinity. So n n goes to infinity that minus one and plus one that we see in the in terms they're going to be negligible once we just think about getting rid of those. What we would end up with is ah, one over in squared. So by construction, this being should, you know, look the same as the AI in terms as n goes to infinity. So what we mean by that is that this ratio should be one. As long as this ratio is something that's finite and non zero, then whatever happens when we sum up, all of these be in terms. We should get the same behavior for summing up all of these in terms. Okay, so one is certainly finite in non zero. And if the sum of all of these being terms from n equals one to infinity, we're going to get convergence. And because this ratio is one, we know that summing up all of these in terms from in equals one to infinity. We should also get convergence

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