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 {n \cos n\pi}{2^n} $

Get the answer to your homework problem.

Try Numerade free for 7 days

Like

Report

Converges

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 5

Alternating Series

Sequences

Series

Campbell 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.

01:48

Test the series for conver…

00:58

0:00

04:49

02:02

test the Siri's for conversions or diversions. Now the first thing we observe here is that it's actually alternating. It might not be obvious right away, but it's actually it's just due to this coastline term because if you were just right, this Siri's out. So the first one you plug in and equals one and then the next term so co sign is just bouncing around between pie into I And that just means that we're getting either one remind us one depending on then. So there's three pie, which is the same thing as co sign a pie again. So we could kind of see the repetition here. So this is just negative. One over two plus two over two square, minus three over to Cube and so on. So, yes, it is alternating. So here we should just take B end for if we want to use the alternating Siri's test, we should define beyond to just be the absolute value. And so this is just an over to to the end, because when we take the absolute value of co sign well, we noticed from our example for writing this out above Don't the co sign is equal to either plus or minus one. So this is just absolute value of plus or minus one, and that's just one. So that's why the one is just and write it out here. So the first condition fall training, Siri says, is that you're bien terms or positive. This is Troop, the second condition is that the limit of being ghost zero. This is also true, and one way to see this. It's just use love. It's a house rule, okay? And there's one more condition to check that the sequence being is decreasing. So me running out of room here will be Write this down here. So one more condition. So let's go out and check this. That's still the question we have to show this. So bien was an over to the end. So the question is whether the Siri's is decreasing so this we can go ahead and show this by because this is equivalent to just and plus one over two of the n plus one less than or equal to and over to the end, you can go ahead and check if this is true. Another option here is to justify enough in terms of B by just replacing and with X. And then we know that f decreases. That's acquittal into the derivative being negative time. So here lets his computer derivative of X using the question rule and then the denominator is always positive. But that numerator, the question is, is whether that's negative. So here this thing will be negative if one minus X, Ellen too, is negative. So that just means one over Ellen, too. Less than X. So if and is bigger than one over natural other too. Then BN is decreasing. Therefore, all three conditions for the alternately Siri's tests or satisfied. So therefore, by alternating Siri's test, the Siri's converges and that's your final answer.

View More Answers From This Book

Find Another Textbook

00:19

Neha bought a fully furnished penthouse for eight crore seventy-five thousan…

01:10

A jewellery shop sells 240 necklaces in a month . 180 of the necklaces were …

Lixin has three ribbons of lengths 160 cm,192 cm and 240 cm. She wishes to c…

01:31

Skill Build 21.Create a circular artwork using the following steps.Step 1: D…

01:27

A bakery sold a total of 3028 coffee buns and blueberry.1560 coffee buns sol…

08:55

Find the cube of 13 by the addition of consecutive odd numbers

01:11

There are 20 coins in a jar each coin is either a nickel or a quarter. If on…

02:11

Find the greatest number that can exactly divide 140, 170 and 155 leaving re…

03:39

express -80/112 as a rational number with numerator -5 and denominator -14