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Problem

Determine whether the sequence converges or diver…

01:05

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Problem 33 Hard Difficulty

Determine whether the sequence converges or diverges. If it converges, find the limit.

$ a_n = \frac {n^2}{\sqrt {n^3 + 4n}} $


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Related Courses

Calculus 2 / BC

Calculus: Early Transcendentals

Chapter 11

Infinite Sequences and Series

Section 1

Sequences

Related Topics

Sequences

Series

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Top Calculus 2 / BC Educators
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Missouri State University

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University of Michigan - Ann Arbor

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Idaho State University

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Boston College

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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
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
Problem 61
Problem 62
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
Problem 93

Video Transcript

okay. We need to look at the Limited and goes to infinity of a m. Figure out whether or not that limit exists and whether or not it's fine. So this is limit as n goes to infinity of in squared over square root of in cubed plus foreign. This problem we do a similar trick to what we've done before As far as we look in the denominator, we looked to see what term in the denominator is going to infinity the fastest except here instead of dividing top and bottom by that term will need to factor out something. So we're gonna need to factor out this in cubed that we have happening here. So factor that out in the denominator and the square root function is multiplication. So we just gotta pull that outside like that. And then we have one plus for over and squared here. So now we can just rewrite this a little bit. Mhm in squared square root of n cubed is in to the 3/2. Mhm. Mhm. Okay, and now we can simplify this part a bit. So this is gonna be in to the Tu minus 3/2 so into the one half. And then we're still dividing by square root of one plus four over n squared. Okay, so hopefully that's legible. Okay, so now if we do that limit on top, divided by limit on bottom, which is something that we're allowed to do as long as we don't get indeterminate form, then we'll see that we're going to get infinity divided by one, which is just gonna be infinity. Yeah, right. So we're allowed to limit on top over limit on bottom as long as we don't get indeterminate form. Indeterminate form would be like infinity over infinity or something divided by zero. If we try doing that here, we're just gonna get infinity over one which is not considered indeterminate form. So that's fine. Okay, So the limit turns out to be infinity, so the limit exists, but it's not finite. So the sequence is still said to diverge

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Related Topics

Sequences

Series

Top Calculus 2 / BC Educators
Catherine Ross

Missouri State University

Kristen Karbon

University of Michigan - Ann Arbor

Michael Jacobsen

Idaho State University

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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