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

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Convergent

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 3

The Integral Test and Estimates of Sums

Sequences

Series

Harvey Mudd College

University of Nottingham

Idaho State University

Boston College

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.

01:02

Determine whether the seri…

01:10

06:39

so to do this, we already have the summation notation. So we're wanting to take the integral. I mean, this is going to be the integral from one to infinity of one over x squared us for the X. This is going to be, um, evaluated when we see the anti derivative of this is the inverse tangent of acts over to We can evaluate that and we end up getting that. This is one half I'm high over to minus are pungent of one house. See that we get the same answer here so we can validate it on DSO. Since this is a finite number, it doesn't go all the way to infinity. That tells us that this is a convergent Siri's.

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