/*! 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 32 Use the Comparison Test or Limit... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 32

Use the Comparison Test or Limit Comparison Test to determine whether the following series converge. $$\sum_{k=1}^{\infty} \sqrt{\frac{k}{k^{3}+1}}$$

Short Answer

Expert verified
Question: Determine if the series $$\sum_{k=1}^{\infty} \sqrt{\frac{k}{k^{3}+1}}$$ converges or diverges. Answer: The Limit Comparison Test with the simpler series $$\sum_{k=1}^{\infty} \frac{1}{k^\frac{3}{2}}$$ did not provide enough information to determine if the original series converges or diverges. Another method or comparison series should be used to determine the convergence of the given series.

Step by step solution

01

Write down the given series and the simpler series to compare with

Our given series is: $$\sum_{k=1}^{\infty} \sqrt{\frac{k}{k^{3}+1}}$$ We will compare this series with the following simpler series: $$\sum_{k=1}^{\infty} \frac{1}{k^\frac{3}{2}}$$
02

Apply the Limit Comparison Test

The Limit Comparison Test requires that we take the limit of the ratio of the terms of the given series (a_k) to the terms of the simpler series (b_k): $$\lim_{k \rightarrow \infty} \frac{a_k}{b_k}$$ Our terms are as follows: $$a_k = \sqrt{\frac{k}{k^{3}+1}}$$ $$b_k = \frac{1}{k^\frac{3}{2}}$$ Now, let's find the limit of the ratio: $$\lim_{k \rightarrow \infty} \frac{\sqrt{\frac{k}{k^{3}+1}}}{\frac{1}{k^\frac{3}{2}}}$$
03

Simplify the expression

Let's simplify the expression within the limit: $$\lim_{k \rightarrow \infty} \frac{k^{\frac{3}{2}}\sqrt{\frac{k}{k^{3}+1}}}$$ Now, let's distribute the k^{\frac{3}{2}} to the expression inside the square root: $$\lim_{k \rightarrow \infty} \sqrt{\frac{k^{4}}{k^{3}+1}}$$
04

Check the result of the limit

As k approaches infinity, the term k^3 dominates the denominator. Therefore, the limit can be simplified as follows: $$\lim_{k \rightarrow \infty} \sqrt{\frac{k^{4}}{k^{3}}} = \lim_{k \rightarrow \infty} \sqrt{k}$$ Since the limit is infinite, the Limit Comparison Test gives no information about the convergence or divergence of the original series. Hence, we need to use another method or comparison series to determine the convergence of the given series.

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

Evaluate the limit of the following sequences. $$a_{n}=\frac{75^{n-1}}{99^{n}}+\frac{5^{n} \sin n}{8^{n}}$$

Determine how many terms of the following convergent series must be summed to be sure that the remainder is less than \(10^{-4} .\) Although you do not need it, the exact value of the series is given in each case. $$\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{(2 k+1) !}$$

Suppose a function \(f\) is defined by the geometric series \(f(x)=\sum_{k=0}^{\infty} x^{2 k}\) a. Evaluate \(f(0), f(0.2), f(0.5), f(1),\) and \(f(1.5),\) if possible. b. What is the domain of \(f ?\)

Evaluate the limit of the following sequences. $$a_{n}=\tan ^{-1}\left(\frac{10 n}{10 n+4}\right)$$

Consider the following infinite series. a. Write out the first four terms of the sequence of partial sums. b. Estimate the limit of \(\left\\{S_{n}\right\\}\) or state that it does not exist. $$\sum_{k=1}^{\infty}(-1)^{k} k$$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

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.