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

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Diverges

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 4

The Comparison Tests

Sequences

Series

Oregon State University

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

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.

01:35

Determine whether the seri…

00:44

03:56

01:07

01:29

03:20

05:52

03:19

let's determine wether this Siri's converges or diversion. Here, let me call this a M, and then let me define Bien to be too over this word of that. So here are all used Limit comparison test. So for Lim comparison test, we look at the limit as n goes to infinity and over being. So let's go ahead and write that out. Here's R A. M so that that's who is not inside the radical and plus two. So that's an over bien and I should have written the limit in the front. So this is the limit is N goes to infinity. Cancel those twos and then here we can go ahead and divide top and bottom by the square event and will end up with one up top and then one plus two over radical and and then, as we take the limit to over radical and ghost zero, and then we're just left over with one now one. It satisfies this inequality. It's bigger than zero, but then it's less than infinity, and this is what's needed. If we want to use limit comparison test, you need a number that's strictly bigger than zero two zero not equal to zero. It's positive, and it cannot equal infinity. Otherwise, you can not use limit comparison test and you have find another method to use. So now t answer our question. We instead compare it with the easier questions. This is the whole point of the limit comparison test. We get to look at the some of the end instead. That's the sum over here. And here you can use the pee test from section eleven point three, and here you have P equals one half because of the square roots. So this will be diversion. And by Lim comparison, test our Siri's will also be diversion because the series that we compared it to the papers and that's our final answer.

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