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Test the series for convergence or divergence.

$ \displaystyle \sum_{n = 2}^{\infty} \frac {1}{n \sqrt{\ln n}} $

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diverges.

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 7

Strategy for Testing Series

Sequences

Series

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

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.

02:51

Test the series for conver…

01:05

0:00

02:08

No, this problem will use the integral test for the interval test. We want for these terms to eventually be strictly decreasing an absolute value to zero. So for that to happen, you'd want for this toe eventually, Just be increasing. So typically speaking, if you know that these things air going toe be going to zero, then you can guess that they're going to be going to zero an absolute value unless you have some trig functions involved. Because with Trig functions you can imagine there being some oscillations occurring. But generally speaking, it's a fairly good gas. If it looks like it's going toe zero, then it's probably going to be going to zero when you throw the absolute value signs around it as well. Like I said, the exception typically being with trig functions, you have to be a little bit more careful with that. Ah, but it is true if you want to be really rigorous and you should look at this, you could take the derivative of this and show that eventually the derivative is going to be positive. The derivative of this is positive for and large enough then that means that eventually this is just going to be decreasing to zero. Case of the integral test is applicable here. So the integral test says that this thing is going to be is going to have the same behavior. Is this integral? Okay, so this is ahh u substitution problem. So the U would be Ellen of N D'You would be won over and Tien So then the integral and consideration turns into Alan of to infinity. You to the minus one half. See you. So this lower bound here, we got that from looking at the lower bound over here. So over here are lower bound was too. And that correspondent to end. So when in is too plugin and has to hear and we get that you is natural log of two. So that's why when we're working with d ur Lord bound his natural log of two Get the upper bound here. We looked to see what's happening with up around here. So hear this corresponds toe in being infinity. Go on in his infinity, you is natural log of infinity, which is still infinity. And then we replaced the one over N d n a with d u. That's what this equation tells us we Khun d'Oh And then we had Then we're left with Do you want over square root of natural log of end, but natural log of in is just you sore left with one over the square root of you which is just you to the minus one half. Okay, so, of course it would be if you're working with D'You You want everything to be written in terms of you. So that's what these equations tell us to dio We know what to do with the d n. We know what to do with the natural lava vent. We we can use these equations to express everything in terms of you. And we can change the bounds by, you know, looking at the bounds appear and figuring out what the bounce correspond to buy this top equation. And then once we have this guy, this is just we could just use the power rule. We get one half, you do one half, and then we're evaluating this from natural log of the two to infinity. And this is certainly going to be something that is infinite. All right. If we plug in infinity here, we're going to get something that blows up. And then we're subtracting what happens when we plug in natural love too. And for you, So have infinity minus something finite. Certainly going to be infinite. Okay, So since this integral turns out to be infinite, the integral test tells us that this son there's also going to be infinite, which means that we get divergence.

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