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Determine whether the sequence converges or diverges. If it converges, find the limit.$ a_n = \cos \left( \frac {n \pi}{n + 1} \right) $

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 1

Sequences

Series

Missouri State University

Harvey Mudd College

University of Michigan - Ann Arbor

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.

03:07

Determine whether the sequ…

02:46

00:26

00:46

to find out whether or not this sequence converges. There diverges. We look at limited n goes to infinity of an If this limit exists and is finally then our sequence is said to converge. Otherwise they're sent to diverge. Since the cosign function is a continuous function were allowed to pull the limit inside of the coastline function. Now we do the trick where we look at the denominator. Look at the term that's going to infinity, the fastest in the denominator, and then divide the top and the bottom by whatever that is. So this case will divide the top and the bottom by N and I will have tie up top and one plus one over in in the denominator. As n goes to infinity, one over in is going to go to zero and we're gonna be left with co sign of pie. Coastline of pi is minus one. So we converge two minus one

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