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Test the series for convergence or divergence.$ \displaystyle \sum_{n = 1}^{\infty} (-1)^n \frac {3n - 1}{2n + 1} $

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 5

Alternating Series

Sequences

Series

Campbell University

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.

04:14

Test the series for conver…

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Let's test the Siri's for convergence or diversions. Now, if we just look at this term over here in this fraction, the women is and goes to infinity ofthree and minus one, two, one plus one. You can use lope. It's hell here to get three halfs and the limit, which is not equal to zero. Therefore, if we continue to multiply this by negative one to the end, that just tells us that for a large and and values negative one to the end. Three and minus one sue one plus one Ossa lates between three halfs and negative three house or US oscillator around east, two numbers. So this just means that the limit of our a end, which is the entire term here limit of an which is the limit as N goes to infinity of negative one to the end three and minus one to one plus one. It does not exist in particular. It's not equal to zero. So the Siri's diverges bye with the author calls the diversions test, and that's the final answer

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