Explanation Given: The series is ∑ k = 2 ∞ a k = ∑ k = 2 ∞ 1 k ( ln k ) p , where p is a real number. Result used: (1)“Suppose f is a continuous, positive, decreasing function, for x ≥ 1 , and let a k = f ( k ) for k = 1 , 2 , 3 , ⋯ then ∑ k = 1 ∞ a k and ∫ 1 ∞ f ( x ) d x either both converge or both diverge. (2) The function f ( x ) is decreasing function if f ′ ( x ) < 0 . (3) The series ∑ n = 2 ∞ 1 k ( ln k ) diverges. Chain rule: d [ f ( x ) ] n d x = n [ f ( x ) ] n − 1 f ′ ( x ) . Product rule: d d x [ f ( x ) g ( x ) ] = d f d x g ( x ) + f ( x ) d g d x . Proof: Compute the value of p if the given series converges. For p = 1 , The series ∑ n = 2 ∞ 1 k ( ln k ) p is diverges by the Result (3). For p ≠ 1 , Consider the function from given series f ( x ) = 1 x ( ln x ) p . The derivative of the function is obtained as follows, f ′ ( x ) = d d x ( 1 x ( ln x ) p ) = d d x ( x − 1 ( ln x ) − p ) Apply the product rule stated above, f ′ ( x ) = ( x − 1 ) d d x ( ( ln x ) − p ) + ( ln x ) − p ⋅ d d x ( x − 1 ) = 1 x ⋅ d d x ( ( ln x ) − p ) + 1 ( ln x ) p ⋅ ( ( − 1 ) x − 1 − 1 ) = 1 x ⋅ d d x ( ( ln x ) − p ) + 1 ( ln x ) p ⋅ ( − x − 2 ) = 1 x ⋅ d d x ( ( ln x ) − p ) − x − 2 ( ln x ) p Apply the chain rule stated above, f ′ ( x ) = 1 x [ ( − p ) ( ln x ) − p − 1 ⋅ d d x ( ln x ) ] − 1 x 2 ( ln x ) p = 1 x [ ( − p ) ( ln x ) − ( p + 1 ) ⋅ 1 x ] − 1 x 2 ( ln x ) p = − p x 2 ( ln x ) p + 1 − 1 x 2 ( ln x ) p = − ( p x 2 ( ln x ) p + 1 + ln x x 2 ( ln x ) p ( ln x ) ) Simplify further and obtain the derivative, f ′ ( x ) = − ( p x 2 ( ln x ) p + 1 + ln x x 2 ( ln x ) p + 1 ) = − p + ln x x 2 ( ln x ) p + 1 Since f ′ ( x ) < 0 for x > e − p , the given function is decreasing by Result (2)

Calculus: Early Transcendentals (2nd Edition)

2nd Edition

ISBN: 9780321947345

Author: William L. Briggs, Lyle Cochran, Bernard Gillett

Publisher: PEARSON

See similar textbooks

Videos

Question

Chapter 8, Problem 60RE

To determine

To show: The series ∑k=2∞1k(lnk)p converges provided p>1.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 8, Problem 59RE

Chapter 8, Problem 61RE

Students have asked these similar questions

The purpose of this problem is to solve the following PDE using a numerical simulation. { af (t, x) + (1 − x)= - Ət af 10²ƒ + მე 2 მე2 = 0 f(ln(2), x) = ex (a) The equation above corresponds to a Feynman-Kac formula. Identify the stochastic process (X)20 and the expectation that would correspond to f(t, x) explicitly. (b) Use a numerical simulation of (X+) above to approximate the values of f(0, x) at 20 discrete points for x, uniformly spaced in the interval [0,2]. Submit a graph of your solution. (c) How would you proceed to estimate the function f(0.1, x). (Briefly explain your method, you do not need to do it.) Extra question: You can explicitly determine the function in (b) (either as a conditional expectation or by solving the PDE). Compare the theoretical answer to your solution.

A sequence is given by the formula an = n/2n^2 +1 . Show the sequence is monotone decreasing for n >1. (Hint: What tool do you know for showing a function is decreasing?)

A sequence is given by the formula an = n 2n2 +1 . Show the sequence is monotone decreasing for n 1. (Hint: What tool do you know for showing a function is decreasing?)

Chapter 8 Solutions

Calculus: Early Transcendentals (2nd Edition)

Ch. 8.1 - Define sequence and give an example.Ch. 8.1 - Suppose the sequence {an} is defined by the...Ch. 8.1 - Suppose the sequence {an} is defined by the...Ch. 8.1 - Prob. 4E Ch. 8.1 - Prob. 5E Ch. 8.1 - Given the series k=1k, evaluate the first four...Ch. 8.1 - The terms of a sequence of partial sums are...Ch. 8.1 - Consider the infinite series k=11k. Evaluate the...Ch. 8.1 - Explicit formulas Write the first four terms of...Ch. 8.1 - Explicit formulas Write the first four terms of...

Additional Math Textbook Solutions

Find more solutions based on key concepts

Birth Length The mean birth length for U.S. children born at full term (after 40 weeks) is 52.2 centimeters (ab...

Introductory Statistics

The simplified expression for the given term

Pre-Algebra Student Edition

Assessment 1-1A How many triangles are in the following figure?

A Problem Solving Approach To Mathematics For Elementary School Teachers (13th Edition)

Whether the requirements for a hypothesis test are satisfied or not.

Elementary Statistics

Identify f as being linear, quadratic, or neither. If f is quadratic, identify the leading coefficient a and ...

College Algebra with Modeling & Visualization (5th Edition)

Type I and Type II Errors. In Exercises 29–32, provide statements that identify the type I error and the type I...

Elementary Statistics (13th Edition)

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

The series ∑ k = 2 ∞ 1 k ( ln k ) p converges provided p > 1 .

Solution Summary: The author explains how the series displaystyle 'underset' converges, provided p>1, where p is a real number.

Videos

To show: The series ∑k=2∞1k(lnk)p converges provided p>1.

Want to see the full answer?

Chapter 8 Solutions

Additional Math Textbook Solutions

The series ∑ k = 2 ∞ 1 k ( ln k ) p converges provided p &gt; 1 .

Solution Summary: The author explains how the series displaystyle 'underset' converges, provided p>1, where p is a real number.

Videos

To show: The series ∑k=2∞1k(lnk)p converges provided p>1.

Want to see the full answer?

Chapter 8 Solutions

Additional Math Textbook Solutions

The series ∑ k = 2 ∞ 1 k ( ln k ) p converges provided p > 1 .