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Prove that if $ \lim_{n \to \infty} a_n = 0 $ and $ \left \{ b_n \right \} $ is bounded, then $ \lim_{n \to\infty} (a_n b_n) = 0. $

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 1

Sequences

Series

Campbell University

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

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.

04:29

$\begin{array}{l}{\text { …

02:08

Prove that if $\lim _{n \r…

so suppose that the limit of a and zero and that being is abounded. Sequels. So this means that there exist, let's say, a positive M David than zero such that, of course here and could even be it could be zero, not a big deal. Absolute value being is always less than or equal to him. So am is a an upper bound for B and positive number here. So here I am is not equal to infinity. Yeah. Now let's use the fact that the limit of a and zero So here, let's just consider any positive number. Absolutely then, since the limit of a and zero there exist and end such that if we take a little and bigger than this big end so such that if we take a little and bigger than begin then a a. And in the absolute value, they're the distance between a and zero is less than Absalon divided by him. So this is just using the definition of limited a zero. So now, if little and is bigger than end, then luscious multiply both sides of this Ibn. So we have an bien minus zero equals just and being an absolute value. This is just a n times being now. This is less than epsilon over m times M. And the reason for this is because on one hand we know that a n an absolute value is less than epsilon over him. On the other hand, we know that BN is less than or equal to him. And if we multiply these out, the EMS cancel and we just haven't Absalon here. And this proves that's a limited of Anne times being a zero by definition, and that's our final answer.

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