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Problem

Use the partial fraction command on your CAS to f…

07:50

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Problem 65 Hard Difficulty

Use the partial fraction command on your CAS to find a convenient expression for the partial sum, and then use this expression to find the sum of the series. Check your answer by using CAS to sum the series directly.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {3n^2 + 3n + 1}{(n^2 + n)^3} $


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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
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Catherine Ross

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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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Watch More Solved Questions in Chapter 11

Problem 1
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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

Video Transcript

let's use a computer algebra system to go ahead and rewrite this term. Here am Yeah. So here in Wolfram Alpha, my computer algebra system. If we scroll down here, we see a partial fraction decomposition that we can work with. So let's go ahead and replace this inside of the Sigma notation. So that's one over and cubed minus one over and plus one cube. This will be much easier to work with and well, actually can use the telescoping method because this is a telescoping series, as we'll see. So here we can rewrite this infinite sum is a limit we're now we're looking at a partial sum from one to k Mhm. So this in the green, including the Sigma, This is R S K R Keith, Partial sum. So using our the computer algebra system, we were able to find a convenient expression for the partial sum. So that's SK, and this is a decent looking expression. And the reason this is convenient is because we can use telescoping method. So now the telescoping method Well, keep writing that limit until the very end. Okay, goes to infinity. So now we start plugging in values of n we start at one, so we have one minus 1/2 cubed. And then we have one over to cube, minus 1/3 cube and so on. And then, on the other hand, we'll have one over K minus one cubed, minus one over K Cube. And then the very last term, When you plug in and equal scare, you have one over K cube minus one over K plus one cubed. Now, before we take that limit, we should go ahead and cancel as much as we can. We can't cancel the one, so the one should still be there in the final answer. However, to over three, we have a negative here. This will cancel with the positive. The 1/3 cubes would have canceled with the next term. And then here we could we see that we could cancel all the way until we get to K Cube. So we're able to cancel all those intermediate terms. However, you still have one more term left with K plus one cubed in the denominator and then finally take that limit. As K goes to infinity, this fraction goes to zero. So the entire sum just goes to one. So before we go log off. The second part of the question is to go ahead and check your answer using a computer algebra system. So here in the third window, I've taken the some from one to infinity, and the sum is also one. So our computer algebra system agrees with our some because we're both getting one, So that's our final answer.

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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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