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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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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 2

Series

Sequences

Missouri State University

Campbell University

Oregon State University

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.

07:50

Use the partial fraction c…

01:57

Use partial fractions to r…

06:34

Use partial fractions to f…

01:07

Use partial fractions to c…

Express the following seri…

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00:58

04:16

01:20

02:02

01:11

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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