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Test the series for convergence or divergence.

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

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 7

Strategy for Testing Series

Sequences

Series

Harvey Mudd College

University of Nottingham

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.

02:51

Test the series for conver…

01:05

04:34

04:48

for this problem. We want to determine if the Siri's is going to converge or diverge on. There's a lot of different tests that we can use this, but in this case, the best one would be an alternating serious test. So with the alternating serious test, what we can do is we can, um, take the derivative of the natural log of X over the square root of X. Um and we see that, but we'll end up getting is that the derivative will be negative for X greater than e squared eso. Now we can take the limit of the natural log of n over the square root of N. I mean, that's the limit as an approaches infinity. So then, using look, he tells rule, we can evaluate this eso we see that one. This is gonna be one over n and down here would be one over to fruit, and I'm going to be equal to okay the limit of to square root of n over and just just, um, one could be zero. That's the limit eso by the alternating Siri's tests. Since that zero, and since the derivative is negative, that's decreasing for large enough value of n We see that by the alternating Siri's test, Um, the given serious that we have will converge.

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