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Determine whether the series is convergent or divergent.$ \displaystyle \sum_{n = 1}^{\infty} \frac {1}{n^2 + 2n + 2} $

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 3

The Integral Test and Estimates of Sums

Sequences

Series

Missouri State University

Oregon State University

Harvey Mudd College

Baylor 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.

06:39

Determine whether the seri…

02:48

02:19

01:06

So we want to determine if this Siri's is going to be convergent or divergent. Um, And to do this, we're going to use the integral. So since we have this summation from one to infinity, we're going to take the integral from one to infinity, and it's going to be integral of one over e squared plus two X plus two Onda. We can also look at that as taking the limit as be approaches infinity from one to be. But doing this, we can use substitution method. Andi, ultimately, they're using substitution method. What we get is that for the integrate this expected X I don't answer is pi over two minus the arc tangent. Okay, we see that we get the same answer. So more importantly, though, we see that this answer does not go to infinity. Rather, it is a finite value. So with that being considered, we know that this Siri's must converge. Andi, that's just by determining how we set up this problem and then evaluating the integral Um, using these calculators is extremely quick and useful. But even if you're doing it by hand, we just recognize how we solve intervals and we see that we'll end up getting the same answer

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