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

Find the values of $ p $ for which the series is convergent.$ \displaystyle \sum_{n = 2}^{\infty} \frac {{1}}{{n(\ln n)^P}} $

Video Answer

Solved by verified expert

This problem has been solved!

Try Numerade free for 7 days

Like

Report

Official textbook answer

Video by Clayton Craig

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 3

The Integral Test and Estimates of Sums

Sequences

Series

Campbell University

Harvey Mudd College

Baylor 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.

09:55

Find the values of $ p $ f…

04:51

04:06

Find the values of $p$ for…

now keep in mind that for and great isn't an equal to two Ellen of n he is going to be possible. So what we have here is a continuous positive, decreasing function. Whatever appears. So that means we can use the integral test. So I'm going to want to integrate, and it's gonna happen. It's when I integrate. I'm going to find the values P that would cause the integral itself converge. So I'm going to do a U substitution of u equals just how on X so that gives you might do just one over x Makes this a nice into girl too. Great. And so I get you know, whatever the end points are of one over you to the P you it becomes, uh, one over negative p plus one. You to the negative people. Nice one. And doing this, they saw that one over you to the P. Is just, you know, the negative p in applied power rule. Of course, this is validated from one end points are but we don't care about you. We care about X. Someone put Our ex is back in. We're going to yet one over negative p plus one an axe, the next vehicles, one evaluated from Tutu. Now we need to take a limit. So we take the limit from era's t Goes to infinity, one over negative peoples one. Hello, next native peoples, one from team Unity. And here I can pull out the one over negative people is one and yeah, trick thi not infinity there. And so get one over negative people. Swon, Lynette asked. He goes to infinity of Ellen T to the negative peoples one minus Ellen too too negative, P plus one. Now I don't care about the Ellen to to the negative people. It's one because it's just a constant. What I want to know is what values of P because this piece to converge. And so to do that, we know that if we have since l nt goes to infinity, you want to make sure that it's on the bottom. So what we need is for a negative p plus one to be Megan. You know, we saw this. You get one less than P. Or in other words, he must be greater than one. So if he is greater than one, the interval converges, meaning that serious convergence

View More Answers From This Book

Find Another Textbook

06:01

Given the region bounded by g (x) = 6 – x^2, the x-axis, x = -1, and x = 2. …

02:39

Estimate the integral below.

03:27

Calculate the following integrals.

02:07

I need urgent help with this multiple choice, calculus questions please!…

02:30

01:29

03:28

01:27

hellp

01:41

About 4% of the population has a particular genetic mutation. 900 people are…

04:30

I need the answers of these two questions ...