   Chapter 11.3, Problem 30E

Chapter
Section
Textbook Problem

Find the values of p for which the series is convergent. ∑ n = 3 ∞ 1 n ln n [ ln ( ln n ) ] p

To determine

To find:

The values of p for which the series n=31nlnnln(lnn)p is convergent

Explanation

1) Concept:

Use the integral test

2) Formula:

Suppose f is a continuous, positive, decreasing function on [1, ] and let an=f(n). Then the series n=1an is convergent if and only if the improper integral 1f(x)dx is convergent.

In other words:

(i) If 1f(x)dx is convergent, then n=1an is convergent

(ii) If 1f(x)dx is divergent, then n=1an is divergent

3) Calculation:

The function fx=1xlnxln(lnx)p is continuous and positive on [3, ]

For p0 function is decreasing on [3, ]

Thus, integral test is applicable

31xlnxln(lnx)pdx=limt3t1xlnxlnlnxpdx for p1

Substitute u=lnlnx so, du=1xlnx

When, x=3, u=ln(

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Use the guidelines of this section to sketch the curve. y=(x3)x

Single Variable Calculus: Early Transcendentals, Volume I

In Exercises 75-98, perform the indicated operations and/or simplify each expression. 88. (3a 4b)2

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

Differentiate. f(x)=xx+cx

Single Variable Calculus: Early Transcendentals 