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Test the series for convergence or divergence.$ -\frac {2}{5} + \frac {4}{6} - \frac {6}{7} + \frac {8}{8} - \frac {10}{9} + \cdot \cdot \cdot $

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divergent

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 5

Alternating Series

Sequences

Series

Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

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.

03:14

Test the series for conver…

02:02

$3-8=$ Test the series for…

00:44

$2-20$ Test the series for…

01:38

01:28

Determine convergence or d…

01:19

01:08

01:03

Use the Integral Test to d…

01:06

02:01

Use the Ratio Test or the …

Let's see whether the Siri's converges are emerges. Now we see that it's alternating. However we see that the numerator increases no sloppy there, let me backtrack. Increases by two, whereas the denominator increases on ly by one so we can actually rewrite this Erie's Yeah, let's say up top. First of all, we should have this negative one to the end power and then up top. We should have two in and then in the denominator, we're adding one each time. But we're starting with an equals one. So four plus end. So here, let's just look at this term here are in. And then let's just go find B End to just be to win over four percent. Now I know that the limit of the end is just equal to two, which is non zero. So this implies that the limit of A M is undefined sense. As n gets really, really large way want will keep all supplying by negative one. But this fraction over here is getting closer to to sew in the limit. A N is getting very close to negative two two negative to two and so on. It's all the limit just will not exist. Therefore, our Siri's we're given above diverges bye, the diversions test, and that's our final answer

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