Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Test the series for convergence or divergence.

$ \displaystyle \sum_{n = 1}^{\infty} \frac {\sin 2n}{1 + 2^n} $

Get the answer to your homework problem.

Try Numerade free for 7 days

Like

Report

Convergent

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 7

Strategy for Testing Series

Sequences

Series

Missouri State University

Oregon State University

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.

00:29

Test the series for conver…

0:00

00:48

00:58

00:38

00:44

01:35

Okay, So, down in the denominator, we have exponential growth and up top. We just have sign of to end so intuitively, this should be clear that this is going to converge and it looks like it's going to converge fast. Looks like it should converge fast enough that it will actually converge. Absolutely. And sometimes it's easier to prove absolute convergence, and it is to just prove regular convergence. And if you are able to prove absolute convergence than that implies regular convergence as well, that's going to be the approach that we're going to take here. So it starts with just the intuition of, you know, assuming that this is going to converge pretty hard, hard enough, Teo, where just converges, absolutely. And in the observation that absolute convergence in this case is going to be something that is easier to show. It'LL be easier to show because when we take the absolute value here, opposition be equal. We should be getting something that's less than one over one plus to the end, right, because sign can't get any bigger than one. And now this is something that looks much nicer to work with, and we can do Ah, another in equality here replaces one plus two to the end with just to the end. Because remember, if we replace the denominator with something smaller, we're going to get something bigger. That's what's happening here. To the end is smaller than one plus two to Ian. But since it's in the denominator, rain it up with something that's larger and then we know that this is well, this is finite. This is finite and all of these terms of positive So all of these terms are positive and we're bounded. So if all of our terms are positive and we're bounded, then it means that we have convergence, okay? And if we have convergence of this thing, well, that means that we have absolute convergence. And if we have absolute convergence, Madam Clise regular convergence Absolute convergence of this guy gives us regular convergence of this guy

View More Answers From This Book

Find Another Textbook

02:25

Evaluate the line integral, where C is the given curve_ Icy ds, C:x=t3 Y =t,…

02:55

We want to fence in & 2700 square foot area next to & building &…

01:55

[-/1 Points]DETAILSTANIAPCAlc9 6.7.015.MY NOTESFind (he amcu…

03:29

Aaecanguler lot I5 to be fenced on two ooposite sidos wilh fencing valued al…

Areciangutar lot is lo be lenced Valued to coposite Sides win {encing value…

02:03

11. Find each of the follwing definite integrals (a) f 3+2d(6) J e"…

03:14

Consider the function f{x,y) = 1Sx2y3Find.f(r;y) dr(b) Find.…

A saiellite which travels around the earth has power supply that provides po…

03:26

Find Hbe A Ma Aecre Analy Evolving Jh peky eaushien_eru_IAe Aa Hub abeul Aht…

03:20

28) [2 Points] Let F = Pi+Qj Then if P and Qvector field on an open and …