/*! 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 29 Use the Comparison Test or Limit... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 29

Use the Comparison Test or Limit Comparison Test to determine whether the following series converge. $$\sum_{k=1}^{\infty} \frac{k^{2}-1}{k^{3}+4}$$

Short Answer

Expert verified
Answer: The series \(\sum_{k=1}^{\infty} \frac{k^{2}-1}{k^{3}+4}\) diverges.

Step by step solution

01

Identify a known convergent or divergent series to compare with our given series

We will compare our given series with the series \(\sum_{k=1}^{\infty} \frac{1}{k}\), a harmonic series known to diverge. This is because the terms in our given series have a similar form to those in the harmonic series, with some higher order terms.
02

Calculate the limit of the ratio of the terms

We will find the limit of the ratio of the terms from the two series as k goes to infinity: $$\lim_{k \to \infty} \frac{a_k}{b_k} = \lim_{k \to \infty} \frac{\frac{k^2-1}{k^3+4}}{\frac{1}{k}}$$ Where \(a_k\) is the terms of our given series and \(b_k\) is the terms of the harmonic series.
03

Simplify the limit

We can simplify the limit expression as follows: $$\lim_{k \to \infty} \frac{k^2-1}{k^3+4} \cdot k = \lim_{k \to \infty} \frac{k^3-k}{k^3+4}$$
04

Calculate the limit

Calculate the limit as k goes to infinity: $$\lim_{k \to \infty} \frac{k^3-k}{k^3+4} = \lim_{k \to \infty} \frac{1 - \frac{1}{k^2}}{1 + \frac{4}{k^3}}$$ As \(k\) goes to infinity, the terms with \(k\) in the denominator go to zero, which means the limit converges to 1.
05

Apply the Limit Comparison Test

Since the limit \(\lim_{k \to \infty} \frac{a_k}{b_k}\) is a positive finite number (in our case, it is 1), the Limit Comparison Test concludes that both the original series and the harmonic series have the same convergence behavior. Since the harmonic series \(\sum_{k=1}^{\infty} \frac{1}{k}\) is a known divergent series, our given series \(\sum_{k=1}^{\infty} \frac{k^{2}-1}{k^{3}+4}\) also diverges.

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

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

Showing that \(\sum_{k=1}^{\infty} \frac{1}{k^{2}}=\frac{\pi^{2}}{6} \operatorname{In} 1734,\) Leonhard Euler informally proved that \(\sum_{k=1}^{\infty} \frac{1}{k^{2}}=\frac{\pi^{2}}{6} .\) An elegant proof is outlined here that uses the inequality $$ \cot ^{2} x<\frac{1}{x^{2}}<1+\cot ^{2} x\left(\text { provided that } 0

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

Use Exercise 89 to determine how many terms of each series are needed so that the partial sum is within \(10^{-6}\) of the value of the series (that is, to ensure \(R_{n}<10^{-6}\) ). a. \(\sum_{k=0}^{\infty} 0.6^{k}\) b. \(\sum_{k=0}^{\infty} 0.15^{k}\)

Consider the following sequences defined by a recurrence relation. Use a calculator, analytical methods, and/or graphing to make a conjecture about the value of the limit or determine that the limit does not exist. $$a_{n+1}=\sqrt{2+a_{n}} ; a_{0}=1, n=0,1,2, \dots$$
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

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.