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Use any test to determine whether the series is absolutely convergent, conditionally convergent, or divergent.

$ \displaystyle \sum_{n = 2}^{\infty} \frac {( - 1)^n}{\ln n} $

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 6

Absolute Convergence and the Ratio and Root Tests

Sequences

Series

Campbell University

Oregon State University

Harvey Mudd College

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.

06:52

Use any test to determine …

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Determine whether the seri…

let's first determine whether the series is absolutely conversion. So we'LL take the absolute value of this term here, and that leaves us with just one over natural log of end. So the question now is whether this new Siri's converges. And so here I would use the comparison test we have won over Natural log of Team. This is excuse me. Natural albumen is bigger than or equal to one over end. And the reason for this is that N is bigger than are equal to, but really bigger than natural log of end and in our case and his two or larger. So this justifies this over here, and we know that the Siri's one over and diversions this's the harmonic series or ego you could also use. The Pee test with P equals one. So since our term up here was bigger than or equal to end, then by the comparison test our series starting at two won over national log of n diverges. So that's using comparison. So here we conclude that it's not absolutely commercial now will determine whether it's conditionally conversion. So this is asking whether or not the original Siri's over here, convergence So you see that this is alternating Siri's. So that suggests we should go ahead and use the alternating Siri's test. Let me go to the next page. So am I. Some here from to the infinity. Negative one and natural law again. Slow fuse alternating Siri's test s t. So here we defined B end to just be the absolute value of this term over here. So that's one over natural log of end and notice that this is bigger than or equal to zero. If Anne is bigger than or equal to two coming from over here the lower bound of the sun. So we have to check a few conditions here. We need that the beings are decreasing. We'LL check that later. Let's first just check that the limit is zero and the denominator goes to infinity. So that limit is zero. We need that to be answered, decreasing. So we would want to show that this is less than or equal to this and this is equivalent to one over natural log and plus one and then we have natural log in. And this is a true statement since we know that natural log of end is less than natural log and plus one. And if you're not convinced why this is true, you could exponentially in each side and you'Ll end up with this. And we definitely know this is true. So we have that the beings are positive that they're limited zero and that the sequence is decreasing. So we we conclude that the Siri's is conditionally conversion.

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