Download the App!

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

Question

Answered step-by-step

Test the series for convergence or divergence.$ \displaystyle \sum_{n = 1}^{\infty} (-1)^n \cos \left( \frac{\pi}{n} \right) $

Video Answer

Solved by verified expert

This problem has been solved!

Try Numerade free for 7 days

Like

Report

Official textbook answer

Video by J Hardin

Numerade Educator

This textbook answer is only visible when subscribed! Please subscribe to view the answer

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 5

Alternating Series

Sequences

Series

Missouri State University

Campbell University

Harvey Mudd College

University of Nottingham

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.

04:59

Test the series for conver…

04:49

02:02

00:58

0:00

01:18

Let's test the Siri's for conversions or diversions. So first, let's look at the limit is N goes to infinity of co sign off Pi over end. A Zen gets really, really large. The fraction will get closer to zero. So we just take the limit. You get coast on zero, and that's one. So this tells us that Well, what's what will happen when we start multiplying by minus one to the end? This just means that the limit is n goes to infinity of an was called this entire thing here this entire term and negative one to the end. Co sign pi a grin. Well, a Zen gets really, really big. We know the co signs getting closer, the one so that just means that this will just continue to oscillate between negative one and one so the limit does not exist. So by the diversions test, and then the reason reusing the test is we've just showed the limit is N goes to infinity. Negative one to the end. Co sign Pi over in is not equal to zero. That's why we're using the diversions tests. Because of the non zero, we can conclude that our Siri's diverges and that's our final answer

View More Answers From This Book

Find Another Textbook

01:23

'The number of fat grams consumed daily by the average American can be …

'In trapezoid LMNO below, median PQ is drawn. If LM = x + 7, ON = 3r + …

02:06

"Mr. Tanner'$ construction class studying the construction of roof…

07:22

'businessman intends to rent car for 3-day business trip The rental is …

02:51

'Michael threw ball into the air from the tOp of building: The height o…

03:05

'A sample of 25 freshman nursing students made a mean score of 77 on a …

07:00

"firm of architects is attempting to get a feel for the sizes ofhouses …

01:37

'Claim: Fewer than 93% of adults have cell phone. In a reputable poll o…

02:36

'Test the claim that the mean GPA of night students is significantly di…