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Show that if $ a_n > 0 $ and $ \lim_{n \to \infty} na_n \not= 0, $ then $ \sum a_n $ is divergent.

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 4

The Comparison Tests

Sequences

Series

Campbell University

Harvey Mudd College

Baylor University

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.

01:10

Show that if $a_{n}>0$ …

00:33

Suppose that $a_{n}>0$ …

00:42

So let's suppose that the limit here and a and A this some number l and were given that it is non zero since the limit evils l This tells us there exist and capital end such that if we take a little and to be larger than this bigon, then the absolute value of any end minus l is less than l over to. So here let's let's note that l is a positive number because it's coming. This limit is just multiplying too. Positive numbers. We're giving the ans positives. So this implies and a and minus l is bigger than over to you. And I could solve this for an now. This tells us that the sum of the AI ends is larger than l or to let me write this one over end. But this series on the right diverges this is just the harmonic series. Or you could even just use the pee test with t equals one. So diverges therefore, by the comparison test this larger Siri's on the left side must also converge. Excuse me. Also diverges No. So we have We just proved that it diverges and that's the final answer

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