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Determine whether the series is convergent or divergent. If it is convergent, find its sum.$ \displaystyle \sum_{n = 1}^{\infty} \frac {2 + n}{1 - 2n} $

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 2

Series

Sequences

Missouri State University

Campbell University

Harvey Mudd College

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.

02:48

Determine whether the seri…

01:32

03:17

06:21

06:39

00:55

Let's determine whether or not the following Syriza's conversion or diversion. If it's convergent, let's go ahead and find some. So looking at this, Siri's here. Let's just focus on the terms that are being added. Usually, the book will denote this by the end. Well, here we can use the with called the Test for Divergence. And in our case, here's the statement. If the limit of a M does not exist or if the limit does exist but equal zero, then the Siri's will diverge. And again, it doesn't matter what your starting point is. It could be any number here instead of one. So on our problem. Let's go ahead and see if we could use this test here. So let's take the limit of an hoops limit as n goes to infinity of a end, that equals limit and goes to infinity two plus and over one minus to end that here, if you like, you could divide top and bottom by N and you have two over and plus one in the numerator, one over and minus two in the denominator. And as we take that limit, thes terms go to zero. But you still have a one on top and negative two on the bottom. That's negative one half, so the limit does exist, but it's not zero. Therefore, this will diverge, diverges and you could even say why Bye, the test for divergence and that's your final answer.

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