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Problem

Determine whether the series converges or diverge…

02:07

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Problem 13 Easy Difficulty

Determine whether the series converges or diverges.
$ \displaystyle \sum_{n= 1}^{\infty} \frac {1 + \cos n}{e^n} $


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Related Courses

Calculus 2 / BC

Calculus: Early Transcendentals

Chapter 11

Infinite Sequences and Series

Section 4

The Comparison Tests

Related Topics

Sequences

Series

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Top Calculus 2 / BC Educators
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Lectures

Video Thumbnail

01:59

Series - Intro

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.

Video Thumbnail

02:28

Sequences - Intro

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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Watch More Solved Questions in Chapter 11

Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
Problem 6
Problem 7
Problem 8
Problem 9
Problem 10
Problem 11
Problem 12
Problem 13
Problem 14
Problem 15
Problem 16
Problem 17
Problem 18
Problem 19
Problem 20
Problem 21
Problem 22
Problem 23
Problem 24
Problem 25
Problem 26
Problem 27
Problem 28
Problem 29
Problem 30
Problem 31
Problem 32
Problem 33
Problem 34
Problem 35
Problem 36
Problem 37
Problem 38
Problem 39
Problem 40
Problem 41
Problem 42
Problem 43
Problem 44
Problem 45
Problem 46

Video Transcript

Let's first note that the terms that we're dealing with their positive. So first, let's look at the one plus co signed in. We know that CO sign of end is less than or equal to one favor than or equal to negative one. So if we add one so all signs of this inequality here we have zero less than or equal to one plus coz I less than or equal to two. So that shows that our numerator is positive and we know he's always positive. The reason I'm checking this is because if you like to use the comparison test, you have to make sure that your Siri's has on ly positive terms. And that's what we have here. A n bigger than zero or equal to That's fine, just no negatives. So now let's go ahead and use comparison his test here. So I know one plus co sign in is less than or equal to two. So this tells me that our Siri's is less than or equal to to overeat of the end. All I'm doing here is just using this inequality that one plus coastline is less than or equal to two and then we can rewrite this. Pull out the two and then we could write. This is one over e to the end. This is a geometric Siri's. We see that our equals one over e rough estimates of this would just be a third. But all that matters is that it's less than one an absolute value, one over three, more or less, and that's less than one. So any time it's geometric series. Satisfied this? We know that it converges. Therefore, since we have a Siri's with positive terms and it's founded above by a convergence here ese by the comparison test our series, which is one plus co sign and over eat of the end. Also convergence okay, and that's your final answer.

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Calculus: Early Transcendentals

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Video Thumbnail

01:59

Series - Intro

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.

Video Thumbnail

02:28

Sequences - Intro

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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