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Test the series for convergence or divergence.$$\sum_{n=1}^{\infty} \frac{1}{n+n \cos ^{2} n}$$

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 7

Strategy for Testing Series

Sequences

Series

Campbell University

University of Michigan - Ann Arbor

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.

0:00

Test the series for conver…

04:59

00:44

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00:54

Yeah. Okay. So here we have the Siri's one over and plus in times close science squared. So for that, we're going to do the comparison test notice that comes on scores less than one. Let's not equal to one. So this is gonna be greater or equal to one over and plus, in times one, and this is greater. This is equals to one half times. The harmonic Siris on this is going to be friendly with goes through feeling, which is which we have something greater than something like infinity. So this is also going to be a divergent series.

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