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

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divergent

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 5

Alternating Series

Sequences

Series

Missouri State University

Oregon State 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.

00:40

Test the series for conver…

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01:42

Let's test the Siri's for convergence or diversions. Now here. Although the Siri's is alternating, I won't use the all trading Siri's tests, and the reason is is I see this term over here, and I noticed that the limit does not go to zero. So that's hints that this Siri's will not converge. So this limit is n goes to infinity. The exponents goes to zero. So we just get either the zero, which is one so non zero. So this means that as n gets large, that are our term. Our end term here a N and the limit will be getting very close to the numbers. Negative one and one be due to the negative one on the outside. And the fact that even the two and goes to zero of excuse me is the one. So this means that the limit as N goes to infinity of A M does not exist in particular. If it doesn't exist, then it cannot be zero. So we conclude at the Siri's diverges, and the reason that it diverges is we have a non zero limited here. So it's the diversions test. Bye. The diversions test always good to explain your reasoning, the same watch, test or theorem you're using Anytime you ever claimed that a Siri's convergence or diverges. So that's is our final answer. Diverges by diversion says.

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