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Determine whether the series converges or diverges.$ \displaystyle \sum_{n = 1}^{\infty} \frac {5 + 2n}{(1 + n^2)^2} $

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 4

The Comparison Tests

Sequences

Series

Missouri State University

Oregon State University

Harvey Mudd College

University of Nottingham

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.

01:16

Determine whether the seri…

01:37

01:20

03:18

01:06

0:00

Determine if the series be…

you can use either the comparison test here or the limit comparison here. Let's just try using the usual that direct comparison test. So this Siri's right here. I couldn't replace this well first, though, actually, let me just keep it as an equal sign. Let me just go to that denominator first and simplify that. So I just have If I just foil that all out, I have to and square plus one plus into the fourth. Now, I could replace this with First of all, let me make the numerator larger. So I know that five is less than or equal to five in. So I'LL just call this five and plus two one and that's up. So if you make a numerator larger, the fraction is a whole good, larger And not only that, I'LL go ahead and make the denominator even smaller by just keeping the large term into the fourth. So here one plus two and swear plus and fourth is bigger than and to the fourth. So I made my denominator smaller and that also makes the fraction is a hole bigger. Therefore, this is why I'm allowed to just use replaced the previous series with the new series, as long as I'm using this inequality. Now let's go ahead and simplify that and equals one to infinity. That gives us seven n over into the fourth. And then you could go ahead and cross off one of those ends on topping on bottom. Your luck with three over on the bottom, and then we just have seven over in cubed. And now this is a P Siri's. So this converges by the Peters with in our case, P equals three. In any time P is larger than one, you always will have conversions. Therefore, also, we should make another observation that we're dealing with positive terms. Therefore, since we're dealing with positive terms, our series converges. Let's not write our let's actually just right the Siri's and equals one to infinity five plus two in all over one plus and swear square converges by the computer sent test, and that's your final answer

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