/*! 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 52 Writing Discuss the convergence ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 52

Writing Discuss the convergence of the sequence \(\left\\{r^{n}\right\\}\) considering the cases \(|r|<1,|r|>1, r=1,\) and \(r=-1\) separately.

Short Answer

Expert verified
Converges to 0 if \(|r|<1\), diverges if \(|r|>1\), converges to 1 if \(r=1\), and diverges if \(r=-1\).

Step by step solution

01

Understanding the Sequence

We are given a sequence \( \{r^n\} \), where \( r \) is a constant and \( n \) is a natural number. The sequence is composed of the terms \( r, r^2, r^3, \ldots \). We need to determine its behavior (convergence) based on different values of \( r \).
02

Case 1 - \(|r|

When \(|r|<1\), as \( n \) becomes very large, the term \( r^n \) becomes very small and approaches zero. Since the terms of the sequence tend towards zero, the sequence converges to zero.
03

Case 2 - \(|r|>1\)

When \(|r|>1\), each successive term of the sequence becomes larger in magnitude than the previous term. As \( n \) increases, \( r^n \) grows without bound, leading to the sequence diverging to infinity or negative infinity, depending on the sign of \( r \).
04

Case 3 - \(r=1\)

If \( r=1 \), every term in the sequence is \( 1^n=1 \), irrespective of \( n \). Thus, the sequence is constantly equal to 1 for all \( n \), and it converges to 1.
05

Case 4 - \(r=-1\)

When \( r=-1 \), the terms of the sequence alternate between \( 1 \) and \( -1 \), i.e., \( (-1)^n \). This alternating nature means the sequence does not settle at a single value, thus it does not converge and is considered divergent.

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Geometric Sequences
A geometric sequence is a sequence of numbers where each term after the first is obtained by multiplying the previous term by a fixed non-zero number called the common ratio, denoted as \( r \). For example, in the sequence \( 2, 4, 8, 16, \ldots \), the common ratio is \( 2 \). The general form of a geometric sequence is \( a, ar, ar^2, ar^3, \ldots \) where \( a \) is the first term and \( r \) is the common ratio.

Key characteristics of geometric sequences include:
  • If \( |r| < 1 \), the terms approach zero as \( n \) increases, making the sequence converge to zero.
  • If \( |r| > 1 \), the absolute value of the terms increases rapidly, leading to divergence. This means the sequence can grow to infinity if \( r > 1 \) or flip signs and diverge if \( r < -1 \).
  • If \( r = 1 \), every term is identical to the first term, leading to the sequence converging to that constant value.
  • If \( r = -1 \), the sequence alternates in sign, which results in divergence since the sequence does not stabilize at a single value.
Sequence Divergence
Sequence divergence occurs when the terms of a sequence do not approach a specific, finite number as the term index \( n \) becomes indefinitely large. Essentially, a divergent sequence either increases or decreases without bound.

There are several patterns of divergence that can occur in sequences:
  • Unbounded Growth: If the terms continue increasing or decreasing to extremely large absolute values, such as in the case where \( |r| > 1 \) in geometric sequences, the sequence diverges to positive or negative infinity.
  • Oscillation: Changes in sign without stabilization, like when \( r = -1 \) in a geometric sequence, can lead to an oscillating pattern where the terms alternate between positive and negative values, preventing convergence to a single number.
Identifying divergence is crucial as it informs us that the sequence does not have a finite limit, impacting how it can be summed or analyzed in calculus and various applications.
Convergent Sequences
A sequence is said to be convergent if its terms get closer and closer to a specific value, known as the limit, as the term index \( n \) approaches infinity. In simple terms, a convergent sequence settles at a single numerical value.

For sequences to converge, certain conditions typically must be met:
  • The terms should steadily decrease in distance from the limit after a certain point, such as when \( |r| < 1 \) in a geometric sequence, causing the terms to shrink towards zero.
  • The sequence could be constant, where all terms equal the same value (e.g., \( r = 1 \) in a geometric sequence), automatically converging as there鈥檚 no deviation from the limit.
Understanding convergence is important in the study of mathematical analysis and calculus because it assures that summing an infinite number of infinitesimally small terms will give a finite total. This principle is core in defining infinite series and understanding their behavior.

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

(a) Find examples to show that if \(\sum a_{k}\) converges, then \(\sum a_{k}^{2}\) may diverge or converge. (b) Find examples to show that if \(\sum a_{k}^{2}\) converges, then \(\quad \sum a_{k}\) may diverge or converge.

Use the Remainder Estimation Theorem to find an interval containing \(x=0\) over which \(f(x)\) can be approximated by \(p(x)\) to three decimal-place accuracy throughout the interval. Check your answer by graphing \(|f(x)-p(x)|\) over the interval you obtained. $$ f(x)=\cos x ; p(x)=1-\frac{x^{2}}{2 !}+\frac{x^{4}}{4 !} $$

Use a graphing utility to confirm that the integral test applies to the series \(\sum_{k=1}^{\infty} k^{2} e^{-k},\) and then determine whether the series converges.

Suppose that the power series \(\sum c_{k}\left(x-x_{0}\right)^{k}\) has radius of convergence \(R\) and \(p\) is a nonzero constant. What can you say about the radius of convergence of the power series \(\sum p c_{k}\left(x-x_{0}\right)^{k}\) ? Explain your reasoning. [Hint: See Theorem \(9.4 .3 .]\)

Use the Remainder Estimation Theorem to find an interval containing \(x=0\) over which \(f(x)\) can be approximated by \(p(x)\) to three decimal-place accuracy throughout the interval. Check your answer by graphing \(|f(x)-p(x)|\) over the interval you obtained. $$ f(x)=\frac{1}{1+x^{2}} ; p(x)=1-x^{2}+x^{4} $$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Decision 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.