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Test the series for convergence or divergence.

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

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 7

Strategy for Testing Series

Sequences

Series

Missouri State University

Harvey Mudd College

University of Michigan - Ann Arbor

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.

00:54

Test the series for conver…

01:06

02:08

01:42

02:13

for this problem. We're going toe rewrite Ellen to the Alan. We'Ll rewrite this natural log in as e to the Ellen of Alan event. Then we're taking that to the power of natural out of end. And then we can rearrange this to write it like e the natural log of n and then take this to the power of Ellen Helin of End. Okay. And again, the exponential function of the log function are in verses of each other. So this is just in. So now we're left with in to the power of natural log natural log of end for in large enough, we know that natural log of natural log of n is eventually going to be larger than to. Because natural log of natural law, Govind is goingto blow up as and goes to infinity. So certainly, at some point it will have to be larger than to Okay. And then this sum there's just the sum that we get when we take our original sum, but then replaced that Ellen of Ellen of n with two. And this thing is going tio converge. So we've shown Is that this thing? If we throw out some finite number of terms, Then we'LL be bounded. Since all of these terms are positive and our Siri's is bounded by something, we know that we must have convergence.

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