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Give an example of a pair of series $ \sum a _n $ and $ \sum b_n $ with positive terms where $ \lim_{n \to \infty} (a_n/b_n) = 0 $ and $ \sum b_n $ diverges, but $ \sum a_n $ converges. (Compare with Exercise 40.)
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Calculus 2 / BC
Chapter 11
Infinite Sequences and Series
Section 4
The Comparison Tests
Sequences
Series
Harvey Mudd College
University of Michigan - Ann Arbor
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.
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Let's give a example of a peer of Siri's for Anne and Bien with positive terms. So Anne and beyond are bigger than zero, and the limit of the fraction must be zero. So will eventually have to deal with this. And then we want that this convergence and that this Siri's diverges so here as an example, let's take being to just be one for each end and let's take and to be won over and square. Then this limit I can rewrite. This is the limit of one over and square over, just one. So that's the limit. One over and square. That's zero so that satisfied. So check also that the are positive terms, so that satisfies this condition. This Siri's converges. So here you can use the pee test with P equals two, and that's bigger than one. So convergence and then the sum of the B in this diverges because it fells the diversions test. What if you take the limit of bien? You just get one, and that's not equal to zero. And that means that the Siri's diverges and so we have all the conditions we want. Let's double check those is positive terms as here, the limit of an over being a zero Check the sum of the being diverges check, and the sum of the conversion is check. It's so this is an example that works, and that's the final answer.
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