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Determine whether the series converges or diverges.$ \displaystyle \sum_{n = 1}^{\infty} \frac {n^2 + n + 1}{n^4 + n^2} $

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Converges

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 4

The Comparison Tests

Sequences

Series

Campbell University

Oregon State University

Baylor University

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.

02:22

Determine whether the seri…

01:16

01:06

01:12

01:05

let's use the Lim comparison test to show that this Siri's will converge. So here this is my and and then let's define being to be and squared over into the fourth, which I can write is one over and swear now the Siri's converges. Why is that so this converges, since it's a P series with P equals two bigger than one. So here, amusing the Pee test from eleven point three the previous section. So Bea and will converge and then we'LL use limit comparison test. So we want to look at this quantity the limit of a n divided by bien And what we're hoping for is that this number satisfies the inequality, basically sees a positive number, not including zero, and it's less than infinity. So it's a real number that's positive. So let's evaluate this. Go ahead and take this limit. So first, let's divide. So we're doing a N, which is over here and then divide this entire expression by one over and swear, since we're dividing by a fraction, that's just multiplying a and over. Bien is just a end times and squared so good and multiply and swear to this numerator we have a into the fourth power and Cube and square and then into the fourth plus and square. Now let's go ahead and divide top and bottom by the highest power that we see in the denominator, which is into the fourth. So on top and bottom. So we're not changing the fraction here because we're doing it to both numerator and denominator. So we have one one over and won over and square and then on the bottom, one plus one over and square. Now go ahead and take that limit. Let and go to Infinity and the numerator. These two terms go to zero and the denominator that's from Ghost zero and we just have one. So C equals one, satisfies this inequality. So by a limit comparison, our Siri's, which was the sum from n equals one to infinity of a M, also converges. And that's a final answer

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