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

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

Oregon State University

University of Michigan - Ann Arbor

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.

06:52

Use any test to determine …

05:08

02:42

03:39

02:45

let's use the root test to see whether the Siri's diverges. So if we call this term here and the root test requires that we look at the end through absolute value and so on our case, let's rewrite this Andrew as the one over and power. And then we'LL have and over natural log of end to the end power. And here there's no need to write the absolute value because and the natural log of end or both positive if Anne is bigger than or equal to two, which is known from our summation, and they're using your laws of exponents. If you raise exponents to another, exponents will multiply those exponents. So here that will give us and over natural log of end. And then in the roux tests, we want to take the limit of this, and if it helps, you can go ahead and replace and with X and use Lopez house rule. So here, go ahead and use the X here. So exes any rule number so we could take a derivative for us before and was just positive image or so the derivative is not defined, so we'd use low Patel's rule here you get one over one over X so Lim X goes to infinity of X. That's infinity, and that's bigger than one. So we conclude that the Siri's diverges by the root test.

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