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Determine whether the series converges or diverges.$ \displaystyle \sum_{n = 1}^{\infty} \sin \left( \frac {1}{n} \right) $

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DIVERGES

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 4

The Comparison Tests

Sequences

Series

Missouri State University

Baylor University

Idaho State 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.

01:17

Determine whether the seri…

03:52

03:02

00:35

02:41

00:49

00:28

let's use the Lim comparison test to determine whether this given series convergence. So if I will use this test, I should call this a M. And then I need some other term being to compare with Ann. And here, just by a little luck, maybe by looking inside of the princess's we just take the end to be won over it. But maybe you're not so lucky. Waited to see why this is a good choice for being, and it gets really, really big sign of one over end and one over and really close together. This is just using the facts that sign and extra approximately equal when x a small, and this could be shown more rigorously. So here, let's look at the limit of a and over bian. That's just the limit as n goes to infinity of sign of one over it over one over. And and instead of simplifying this, let's just create another letter here. Let's call them to the one over, and that I can replace this notice that as n goes to infinity, that's a global in tow. M going to zero because one over infinity a zero and then I could write this a sign over and and this limit, which you might remember from your taking derivatives and limits You could use local tiles rule here, but sine X over X goes toe one in the limit. Therefore, we can use the Lim comparison test. So instead of looking at our Siri's, we just have to look at this series here, and we know the serious that verges. This is the well known harmonic series, or you could just say that it diverges because it's a piece. Areas with P equals one. So by the pee test, this is from eleven point three with p equals one, therefore by Lim comparison, which we're allowed to use because we have a positive number. That's so first of all, it's bigger than zero and two. It's not infinite part of me. So by Lim comparison, our Siri's also converges deceives me. Also diverges, diverges, and that's our final answer. Why limit comparison

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