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A series $ \sum a_n $ is defined by the equations

$ a_1 = 1 $ $ a_{n+1} = \frac {2 + \cos n}{\sqrt{n}} a_n $

Determine whether $ \sum a_n $ converges or diverges.

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 6

Absolute Convergence and the Ratio and Root Tests

Sequences

Series

Campbell University

Oregon State University

Harvey Mudd 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.

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here were given a sequence whose first term is one and for end bigger than or equal to one earned. Let's say, two. The end term is given by this expression here, and we'd like to know whether or not the Siri's a end conversions. So let's apply on this one. Let's go for the ratio test. So that requires us to look at and plus one over n an absolute value. No. So using our formula here, we can write the numerator, and then we still have our hand on the denominator. And fortunately, those will cancel, giving us two plus co sign and over Rouen. And we can drop the absolute value because all the terms to plus coastlines positive also because co signs between negative one and one that implies two plus co signs between one and three. And so when we take the limit is n goes to infinity. The numerator will be varying between one and three, but the denominator goes to infinity so that women will be zero. That's less than one. So that means the Siri's converges by the ratio test. Okay,

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