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Determine whether the series converges or diverges.$ \displaystyle \sum_{k = 1}^{\infty} \frac {\ln k}{k} $
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01:00
Yiming Zhang
Calculus 2 / BC
Chapter 11
Infinite Sequences and Series
Section 4
The Comparison Tests
Sequences
Series
Campbell University
University of Michigan - Ann Arbor
University of Nottingham
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.
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Determine whether the seri…
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Determine whether each ser…
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Hello. So here we have these somewhere we have K. Going from one to infinity of the natural log of K over K. So um well we can expand this series with expect to K. So therefore we have while the natural log of 1/1 plus the natural log of 2/2 plus well and so on. Plus we get these some where we get, let's say K. Now goes from three to infinity of the natural log of K over K. So we then take the series. Here we have the K. E goes from three to infinity of Ellen K over K. And we're going to use the comparison test to determine whether the series converges or diverges. So we know that the comparison test right? If we take the some of B seven is convergent where we get that A seven is less than equal to B. Seven for all end than the sum of a servant is also convergent. And if the Saban is out of some of the seven is divergent and we get a servant is greater than or equal to B. Seven for all N. Than the sum of a servant is also divergent. So here, since we have that Ellen K is greater than one for all K, greater than equal to three, it follows that we have the natural log of K over K is going to be greater than one over K for all K greater than or equal to three. So here let's let um a sub K equals the natural log of K over K. And let's let be sub K equal are one over K. Now, since each term of the series um some going from K equals three to infinity of L. N. F. K over K is greater than the corresponding terms of the series where K goes from three to infinity of one over K. Which we have as a P series with P equal to one, which is divergent. Therefore, we had at the series K, going from three to infinity of Ellen of K over K must be divergent as well, so must be divergent by the comparison test.
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