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Problem

Determine whether the series is convergent or div…

06:27

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Problem 42 Medium Difficulty

Determine whether the series is convergent or divergent. If it is convergent, find its sum.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {e^n}{n^2} $


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Related Courses

Calculus 2 / BC

Calculus: Early Transcendentals

Chapter 11

Infinite Sequences and Series

Section 2

Series

Related Topics

Sequences

Series

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Top Calculus 2 / BC Educators
Grace He
Catherine Ross

Missouri State University

Kristen Karbon

University of Michigan - Ann Arbor

Joseph Lentino

Boston College

Calculus 2 / BC Courses

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
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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
Problem 47
Problem 48
Problem 49
Problem 50
Problem 51
Problem 52
Problem 53
Problem 54
Problem 55
Problem 56
Problem 57
Problem 58
Problem 59
Problem 60
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Problem 63
Problem 64
Problem 65
Problem 66
Problem 67
Problem 68
Problem 69
Problem 70
Problem 71
Problem 72
Problem 73
Problem 74
Problem 75
Problem 76
Problem 77
Problem 78
Problem 79
Problem 80
Problem 81
Problem 82
Problem 83
Problem 84
Problem 85
Problem 86
Problem 87
Problem 88
Problem 89
Problem 90
Problem 91
Problem 92

Video Transcript

let's go ahead and determine whether or not the serious conversions and if it's convergent, will go ahead and find the summer as well. So for this problem here, the first thing we might look at to see is if it's geometric and in this one we can just we could see that the answer is no. You cannot write this problem in the form. Eight times are the end You were A and R don't depend on him. So let's try another test here. Let's try the test for diverges, which is in the which is stated in the section in the book. So here we look at the women and goes to infinity of either the end over and square. Now I claim that this is will be equal to infinity and a way to see this hour. This technically the limit. Right now, the ways rent is of the form infinity over infinity and from complice. You might remember this and your previous calculus course. This is what the author calls the indeterminate form and usually use low Patel's rule, which is perfectly fine here. But some people would not use Low Patel when you're using the value of that because these functions here or not continuous because you're not using all roll numbers, you're only using natural numbers. So what some people would do is they would love replace and with X That way you're dealing with the rial valued functions here and these are continuous these air differential so you can go ahead and use love with house rules. So let me just write. I'Ll hop there for low Patel. We take the limit X goes to infinity derivative on the top to rivet upon the bottle. This is still of the forum infinity over infinity, So use low Patel again. So take the derivative of top and bottom once more and then we get either the ex over to. And then now this is either the X divided by a number and this is equal to infinity. So therefore we have that the limit exist. So let's just write it this way. Since the limit of N goes to infinity of either then over and square is not equal to zero. We read it this way equals infinity. So it exists but is not equal to zero. The Siri's and equals want to infinity E n over and square diverges by the diversions test. This is a very important test, and that's my final answer.

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Related Topics

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Top Calculus 2 / BC Educators
Grace He

Numerade Educator

Catherine Ross

Missouri State University

Kristen Karbon

University of Michigan - Ann Arbor

Joseph Lentino

Boston College

Calculus 2 / BC Courses

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.

Join Course
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