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Calculate the sum of the series $ \sum_{n = 1}^{\infty} a_n $ whose partial sums are given.$ s_n = \frac {n^2 - 1}{4n^2 + 1} $

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 2

Series

Sequences

Missouri State University

University of Michigan - Ann Arbor

University of Nottingham

Idaho 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.

01:44

Calculate the sum of the s…

00:59

Calculate the $N^{\text {t…

04:47

Find the sum of each serie…

01:27

01:45

Find the sum of the given …

00:54

Find the indicated partial…

02:33

Express the following seri…

01:15

Find the sum of the series…

04:58

01:18

00:56

Find sum of the series su…

Let's calculate the sum of the Siri's here. So by definition, first lesson's rewrite this as a limit. Let's actually is a different letters since and is being used. Let's say our goes to infinity and then we have our son from an equals one of Tau are And then, by definition, I could go ahead and replace this with S R. This is the Arth partial some This is the notation at the author uses in the section for the partial sums as his talent. It stands for the partial Some If this sub script are is telling you how many terms of this a sequence that you want to add up And if you started a one and you always go up to the index here, so up they are. So this is just limit as our goes to infinity of S r. And then this is where we can use our formula for the esses. So here we're just using our instead of end again. The only reason I'm not using and here under this is because they decided to already use and over here, so replace as and are as our with R squared minus one over four R squared plus one. Now, you may recall many several ways to go around about this limit here, so you will want to be familiar with your limits. You could do open house rule here because the numerator and denominator we're both going to infinity. So if you'd like or this case, you could actually just go ahead and divide. It's happened bottom by the highest power of armed, the bottom civil divides happened. Bottom by R squared so top. That's what we give on the bottom. That's what we get. And as we take the limit, our goes to infinity. Those go to zero and we're left with one over four, and that's your final answer.

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