/*! 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 2 Is it true that if the terms of ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 2

Is it true that if the terms of a series of positive terms decrease to zero, then the series converges? Explain using an example.

Short Answer

Expert verified
Based on the analysis and the example of the harmonic series, we can conclude that the given statement is false. A series with positive terms decreasing to zero does not necessarily ensure convergence. The harmonic series is an example where the terms decrease to zero, but the series itself diverges.

Step by step solution

01

Understanding series convergence

In order for a series to converge, the sum of the terms must approach a finite limit. In other words, the sum of the terms should not go to infinity. If the sum approaches infinity, the series is said to diverge.
02

Analyzing the statement

The given statement claims that if the terms of a series of positive terms decrease to zero, then the series converges. Our goal is to find an example to verify the validity of the statement.
03

Presenting an example

One of the classic examples that can be used to verify the statement is the harmonic series. The harmonic series is defined as: \[ \sum_{n=1}^{\infty} \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots \] Notice that the terms of the series are positive and decrease to zero. According to the given statement, this series should converge.
04

Assessing convergence

To verify if the series converges or not, we will use the comparison test. The comparison test states that if a smaller series diverges, then, any larger series will also diverge. Let us compare the harmonic series with another series as follows: \[ 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots > 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{4} + \cdots \] Observe that the second series consists of summing up the powers of 2. The second series can be written as: \[ 1 + \frac{1}{2} + (\frac{1}{4}+\frac{1}{4})+(\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8})+ \cdots \] This series can be seen as a geometric series with a common ratio of 1/2: \[ 1 + \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \cdots \] This geometric series diverges since it does not have a finite limit. The limit is infinity as the number of terms approaches infinity. Therefore, by the comparison test, the harmonic series also diverges.
05

Conclusion

The harmonic series is an example of a series with positive terms that decrease to zero but diverge instead of converging. Therefore, the given statement is false. A series with terms decreasing to zero does not necessarily guarantee that the series will converge.

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

Determine whether the following statements are true and give an explanation or counterexample. a. A series that converges must converge absolutely. b. A series that converges absolutely must converge. c. A series that converges conditionally must converge. d. If \(\sum a_{k}\) diverges, then \(\Sigma\left|a_{k}\right|\) diverges. e. If \(\sum a_{k}^{2}\) converges, then \(\sum a_{k}\) converges. f. If \(a_{k}>0\) and \(\sum a_{k}\) converges, then \(\Sigma a_{k}^{2}\) converges. g. If \(\Sigma a_{k}\) converges conditionally, then \(\Sigma\left|a_{k}\right|\) diverges.

Determine whether the following series converge absolutely or conditionally, or diverge. $$\sum_{k=1}^{\infty} \frac{(-1)^{k}}{k^{2 / 3}}$$

The CORDIC (COordinate Rotation DIgital Calculation) algorithm is used by most calculators to evaluate trigonometric and logarithmic functions. An important number in the CORDIC algorithm, called the aggregate constant, is \(\prod_{n=0}^{\infty} \frac{2^{n}}{\sqrt{1+2^{2 n}}},\) where \(\prod_{n=0}^{N} a_{n}\) represents the product \(a_{0} \cdot a_{1} \cdots a_{N}\). This infinite product is the limit of the sequence $$\left\\{\prod_{n=0}^{0} \frac{2^{n}}{\sqrt{1+2^{2 n}}} \cdot \prod_{n=0}^{1} \frac{2^{n}}{\sqrt{1+2^{2 n}}}, \prod_{n=0}^{2} \frac{2^{n}}{\sqrt{1+2^{2 n}}} \ldots .\right\\}.$$ Estimate the value of the aggregate constant.

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

Determine whether the following series converge or diverge. $$\sum_{k=2}^{\infty} \frac{4}{k \ln ^{2} k}$$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Mechanics Maths

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.