/*! 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 77 Let \(\left\\{a_{n}\right\\}\) b... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 77

Let \(\left\\{a_{n}\right\\}\) be an increasing sequence such that \(2 \leq a_{n} \leq 4\). Explain why \(\left\\{a_{n}\right\\}\) has a limit. What can you conclude about the limit?

Short Answer

Expert verified
By applying the Monotone Sequence Theorem, we can conclude that the sequence \(\left\{a_{n}\right\}\) has a limit. Furthermore, the limit of this sequence is confined to the interval [2, 4].

Step by step solution

01

Recognize the Properties of the Sequence

The given sequence, \(\left\{a_{n}\right\}\), is increasing and bounded by two numbers, 2 and 4. In other words, for any n, we have \(2 \leq a_{n} \leq 4 \).
02

Apply the Monotone Sequence Theorem

The Monotone Sequence Theorem states that every bounded and increasing (or decreasing) sequence has a limit. Since our sequence is increasing and bounded, we can conclude that it has a limit.
03

Deduce the Possible Values of the Limit

Given the nature of the sequence (increasing and bounded), the limit of such sequence must lie in the interval [2,4]. Therefore, the limit of the sequence \(a_{n}\), if it exists, must be a number L satisfying \(2\le L \le 4\). This is derived from what's called the squeeze theorem or sandwich theorem, which states that if a sequence is squeezed between two other sequences that converge to the same limit, then that sequence also converges to that limit.

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

Consider the sequence \(\left\\{a_{n}\right\\}=\left\\{\frac{1}{n} \sum_{k=1}^{n} \frac{1}{1+(k / n)}\right\\}\). (a) Write the first five terms of \(\left\\{a_{n}\right\\}\) (b) Show that \(\lim _{n \rightarrow \infty} a_{n}=\ln 2\) by interpreting \(a_{n}\) as a Riemann sum of a definite integral.

In Exercises 81-84, give an example of a sequence satisfying the condition or explain why no such sequence exists. (Examples are not unique.) A monotonically increasing sequence that converges to 10

In Exercises \(53-68,\) determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \frac{n+10}{10 n+1} $$

Find the sum of the convergent series. $$ \sum_{n=0}^{\infty} 2\left(-\frac{2}{3}\right)^{n} $$

Describe the difference between \(\lim _{n \rightarrow \infty} a_{n}=5\) and \(\sum_{n=1}^{\infty} a_{n}=5\).
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

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.