/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 38 Use the Comparison Test or Limit... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 38

Use the Comparison Test or Limit Comparison Test to determine whether the following series converge. $$\sum_{k=2}^{\infty} \frac{1}{(k \ln k)^{2}}$$

Short Answer

Expert verified
Answer: The given series \(\sum_{k=2}^{\infty} \frac{1}{(k \ln k)^{2}}\) converges.

Step by step solution

01

Identify the given series and a series to compare it with

We are given the series \(\sum_{k=2}^{\infty} \frac{1}{(k \ln k)^{2}}\). Notice that it is similar to the p-series \(\sum_{n=2}^{\infty} \frac{1}{n^2}\) where \(p = 2\). So, we can compare our given series with this p-series to determine convergence.
02

Check the positivity of the series terms

Both the given series and the p-series have positive terms as \(k\) ranges from 2 to \(\infty\), and the natural logarithm function is positive for \(k > 1\). So, we can proceed with the Comparison Test or Limit Comparison Test.
03

Apply the Limit Comparison Test

We will use the Limit Comparison Test to determine the convergence of the given series. Let \(a_k = \frac{1}{(k \ln k)^2}\) and \(b_k = \frac{1}{k^2}\). We must compute the limit: $$\lim_{k \to \infty} \frac{a_k}{b_k} = \lim_{k \to \infty} \frac{\frac{1}{(k \ln k)^2}}{\frac{1}{k^2}} = \lim_{k \to \infty} \frac{k^2}{(k \ln k)^2}$$
04

Simplify the limit

To simplify the limit, divide the numerator and the denominator by \(k^2\): $$\lim_{k \to \infty} \frac{k^2}{(k \ln k)^2} = \lim_{k \to \infty} \frac{1}{(\ln k)^2}$$
05

Evaluate the limit

As \(k\) goes to infinity, the natural logarithm of \(k\) also goes to infinity. Therefore, \((\ln k)^2\) goes to infinity, and the limit becomes: $$\lim_{k \to \infty} \frac{1}{(\ln k)^2} = 0$$
06

Use the Limit Comparison Test result

Because the limit is a finite positive value (0 in this case), the Limit Comparison Test tells us that both series have the same convergence behavior. Since the p-series \(\sum_{n=2}^{\infty} \frac{1}{n^2}\) is a convergent series (as \(p = 2 > 1\)), it follows that the given series \(\sum_{k=2}^{\infty} \frac{1}{(k \ln k)^{2}}\) also converges.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The expression where the process continues indefinitely, is called a continued fraction. a. Show that this expression can be built in steps using the recurrence relation \(a_{0}=1, a_{n+1}=1+1 / a_{n},\) for \(n=0,1,2,3, \ldots . .\) Explain why the value of the expression can be interpreted as \(\lim a_{n}\). b. Evaluate the first five terms of the sequence \(\left\\{a_{n}\right\\}\). c. Using computation and/or graphing, estimate the limit of the sequence. d. Assuming the limit exists, use the method of Example 5 to determine the limit exactly. Compare your estimate with \((1+\sqrt{5}) / 2,\) a number known as the golden mean. e. Assuming the limit exists, use the same ideas to determine the value of where \(a\) and \(b\) are positive real numbers.

Consider the sequence \(\left\\{x_{n}\right\\}\) defined for \(n=1,2,3, \ldots\) by $$x_{n}=\sum_{k=n+1}^{2 n} \frac{1}{k}=\frac{1}{n+1}+\frac{1}{n+2}+\dots+\frac{1}{2 n}$$ a. Write out the terms \(x_{1}, x_{2}, x_{3}\) b. Show that \(\frac{1}{2} \leq x_{n}<1,\) for \(n=1,2,3, \ldots\) c. Show that \(x_{n}\) is the right Riemann sum for \(\int_{1}^{2} \frac{d x}{x}\) using \(n\) subintervals. d. Conclude that \(\lim _{n \rightarrow \infty} x_{n}=\ln 2\)

Two sine series Determine whether the following series converge. a. \(\sum_{k=1}^{\infty} \sin \frac{1}{k}\) b. \(\sum_{k=1}^{\infty} \frac{1}{k} \sin \frac{1}{k}\)

Consider the following situations that generate a sequence. a. Write out the first five terms of the sequence. b. Find an explicit formula for the terms of the sequence. c. Find a recurrence relation that generates the sequence. d. Using a calculator or a graphing utility, estimate the limit of the sequence or state that it does not exist. A material transmutes \(50 \%\) of its mass to another element every 10 years due to radioactive decay. Let \(M_{n}\) be the mass of the radioactive material at the end of the \(n\) th decade, where the initial mass of the material is \(M_{0}=20 \mathrm{g}.\)

Infinite products Use the ideas of Exercise 88 to evaluate the following infinite products. $$\text { a. } \prod_{k=0}^{\infty} e^{1 / 2^{k}}=e \cdot e^{1 / 2} \cdot e^{1 / 4} \cdot e^{1 / 8} \ldots$$ $$\text { b. } \prod_{k=2}^{\infty}\left(1-\frac{1}{k}\right)=\frac{1}{2} \cdot \frac{2}{3} \cdot \frac{3}{4} \cdot \frac{4}{5} \dots$$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.