Download the App!

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

Sent to:
Search glass icon
  • Login
  • Textbooks
  • Ask our Educators
  • Study Tools
    Study Groups Bootcamps Quizzes AI Tutor iOS Student App Android Student App StudyParty
  • For Educators
    Become an educator Educator app for iPad Our educators
  • For Schools

Problem

Show that series $ \sum ( - 1)^{n-1}b_n, $ where …

06:13

Question

Answered step-by-step

Problem 34 Hard Difficulty

For what value of $ p $ is each series convergent?

$ \displaystyle \sum_{n = 2}^{\infty} ( - 1)^{n-1} \frac {(\ln n)^P}{n} $


Video Answer

Solved by verified expert

preview
Numerade Logo

This problem has been solved!

Try Numerade free for 7 days

JH
J Hardin
Numerade Educator

Like

Report

Textbook Answer

Official textbook answer

Video by J Hardin

Numerade Educator

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

Related Courses

Calculus 2 / BC

Calculus: Early Transcendentals

Chapter 11

Infinite Sequences and Series

Section 5

Alternating Series

Related Topics

Sequences

Series

Discussion

You must be signed in to discuss.
Top Calculus 2 / BC Educators
Catherine Ross

Missouri State University

Kristen Karbon

University of Michigan - Ann Arbor

Samuel Hannah

University of Nottingham

Michael Jacobsen

Idaho State University

Calculus 2 / BC Courses

Lectures

Video Thumbnail

01:59

Series - Intro

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.

Video Thumbnail

02:28

Sequences - Intro

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.

Join Course
Recommended Videos

04:43

For what value of $ p $ i…

06:41

For what value of $ p $ i…

03:36

For what values of $p$ is …

02:45

For what values of $p$ is …

01:03

For what values of $p$ is …

02:16

For what values of $p$ is …

07:56

For what values of $ p $ d…

06:20

for what values of p does …

Watch More Solved Questions in Chapter 11

Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
Problem 6
Problem 7
Problem 8
Problem 9
Problem 10
Problem 11
Problem 12
Problem 13
Problem 14
Problem 15
Problem 16
Problem 17
Problem 18
Problem 19
Problem 20
Problem 21
Problem 22
Problem 23
Problem 24
Problem 25
Problem 26
Problem 27
Problem 28
Problem 29
Problem 30
Problem 31
Problem 32
Problem 33
Problem 34
Problem 35
Problem 36

Video Transcript

Let's find the values of P for which the syriza's conversion. So here we see that this Siri's is alternating. So let's just go ahead and not make any assumptions yet about Pete. But it could be any real number. And let's see how far we can go with the alternating series test. Perhaps this will converse for any number of pee. So here we defined the end to just be the positive part of this and term here. So issues natural log of end to the P over. And so here. This is bigger than early or actually here. This will actually just being bigger than zero if and is bigger than or equal to two. That's the assumption over here. So actually, I take that back. So here we want and to be bigger than three. All that matters is that it's eventually true. Doesn't have to be true for every end. But there has to be a point in which we hit a number in this case three and every number. After that, it's positive. So then that's our BM. That's the first condition is that it has to be positive. The second condition is that we need the limit as n goes to infinity of being to be zero. So that's natural log of end to the P over end and here what I would do is if I want to use Lopez's house rule. I'Ll just use x So here by low Patel this limit So here really depends on peace off p zero. So first, if if peace equals is you know or even less than zero, then the limit well equal zero. Otherwise it peace positive then it's not so obvious. We have infinity over infinity Then the limit is of the form infinity over infinity. And this is where we use Lopez house rule So we would take the derivative of the numerator And then we get Exxon the denominator from the General And then we take the derivative of the denominator ex specious one. And if we could, then we could look at T minus one. If p minus one is less than or equal to zero, the limit is is equal to zero. Otherwise it appears bigger. P minus one is bigger than zero. I want to use Lopez house rule again and we'Ll continue to keep subtracting one from the P Eventually this will be zero or negative and then we'Ll still have an excellent bottom from the chain rule and this limit will end up being zero. So here I'll summarize that limit of being equal zero and the last condition to check is that BN is eventually decreasing. So let's go to the next page. This was our b bien. So the third condition in the final one for altering Siri's is that the sequence is decreasing eventually, Not not necessarily for all in so there. To show this, we could define this function f and based on the end. And then I know that if that prime is negative, that's equivalent Teo being decreasing. So this is what we're gonna put a question mark here because that's what we want to check. F is decreasing That's equivalent to the sequence being decreasing so we can use this method a line in the book. Teo, check whether beings decreasing witches have to take the derivative of this eh function. So I used the question rule. So here you'll get P l n x if he minus one over x ten times the denominator and then minus l a next to the P times one over denominators, Claire. So this Khun simplify you pull out a Ellen to the P minus one power. So here, if X is bigger than her equal to three, then Ellen of X is positive. I know the denominators positive. The only question is is whether P minus Elena X is positive. If this thing is negative, this means that the entire fraction is negative. And this will be true when piers less than Ellen X, which is equivalent to even the pee less than X. So this is eventually true when X increases so if and is bigger than me to the P, then be n plus one is less than or equal to be in. So this bee in sequence is eventually decreasing. And that's what we wanted increasing. And let me summarize on the last page here by the alternating Siri's test, the Siri's convergence. And now we tio answer the original question. Since we made no assumptions about P, we've shown that the Siri's converges for Alti converges for any really number team, and that's our final answer

Get More Help with this Textbook
James Stewart

Calculus: Early Transcendentals

View More Answers From This Book

Find Another Textbook

Study Groups
Study with other students and unlock Numerade solutions for free.
Math (Geometry, Algebra I and II) with Nancy
Arrow icon
Participants icon
85
Hosted by: Ay?Enur Çal???R
Math (Algebra 2 & AP Calculus AB) with Yovanny
Arrow icon
Participants icon
53
Hosted by: Alonso M
See More

Related Topics

Sequences

Series

Top Calculus 2 / BC Educators
Catherine Ross

Missouri State University

Kristen Karbon

University of Michigan - Ann Arbor

Samuel Hannah

University of Nottingham

Michael Jacobsen

Idaho State University

Calculus 2 / BC Courses

Lectures

Video Thumbnail

01:59

Series - Intro

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.

Video Thumbnail

02:28

Sequences - Intro

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.

Join Course
Recommended Videos

04:43

For what value of $ p $ is each series convergent? $ \displaystyle \sum_{n …

06:41

For what value of $ p $ is each series convergent? $$ \displaystyle \sum_{n = …

03:36

For what values of $p$ is each series convergent? $$\sum_{n=2}^{\infty}(-1)^{n…

02:45

For what values of $p$ is each series convergent? $$\sum_{n=1}^{\infty} \frac{…

01:03

For what values of $p$ is the following series convergent? $$\sum_{n=1}^{\in…

02:16

For what values of $p$ is each series convergent? $\sum_{v=2}^{\infty}(-1)^{n-…

07:56

For what values of $ p $ does the series $ \sum_{n = 2}^{\infty} 1/(n^P \ln n) …

06:20

for what values of p does the series summation n=2 to infinity 1/(n^p ln(n)) co…
Additional Mathematics Questions

01:32

"Factorise
(x+y+z)^3-x^3-y^3-z^3
(16)
Factorise the expression…

03:58

'prove that :- tanA/1-cotA + cotA/1-tanA = secAcosecA+1'

Add To Playlist

Hmmm, doesn't seem like you have any playlists. Please add your first playlist.

Create a New Playlist

`

Share Question

Copy Link

OR

Enter Friends' Emails

Report Question

Get 24/7 study help with our app

 

Available on iOS and Android

About
  • Our Story
  • Careers
  • Our Educators
  • Numerade Blog
Browse
  • Bootcamps
  • Books
  • Notes & Exams NEW
  • Topics
  • Test Prep
  • Ask Directory
  • Online Tutors
  • Tutors Near Me
Support
  • Help
  • Privacy Policy
  • Terms of Service
Get started