/*! 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 81 Which of the sequences \(\left\\... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 81

Which of the sequences \(\left\\{a_{n}\right\\}\) converge, and which diverge? Find the limit of each convergent sequence. \(a_{n}=\tan ^{-1} n\)

Short Answer

Expert verified
The sequence converges to \(\frac{\pi}{2}\).

Step by step solution

01

Understand the Sequence

The sequence given is \(a_n = \tan^{-1} n\). This means that for each \(n\), \(a_n\) is the arctangent of \(n\). The question asks whether this sequence converges or diverges and to find the limit if it converges.
02

Consider the Behavior at Infinity

Analyze the behavior of \(\tan^{-1} n\) as \(n\) approaches infinity. The arctangent function, \(\tan^{-1} x\), approaches \(\frac{\pi}{2}\) as \(x\) becomes very large. Therefore, \(a_n = \tan^{-1} n\) should approach \(\frac{\pi}{2}\) as \(n\) goes to infinity.
03

Determine Convergence or Divergence

Since \(a_n = \tan^{-1} n\) approaches \(\frac{\pi}{2}\) as \(n\) increases, the sequence \(a_n\) converges. A sequence converges if it approaches a particular value as \(n\) becomes very large.
04

Identify the Limit

Since the sequence \(a_n\) converges, we determine its limit. As \(n \to \infty\), \(a_n\) approaches \(\frac{\pi}{2}\). Thus, the limit of the sequence \(a_n = \tan^{-1} n\) is \(\frac{\pi}{2}\).

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.

Arctangent Function Behavior
The arctangent function (denoted as \( \tan^{-1} x \) or \( \arctan(x) \)) is essential when examining the sequence \( a_n = \tan^{-1} n \). This function maps real numbers into the interval \((-\frac{\pi}{2}, \frac{\pi}{2})\). Here are some crucial behaviors of the arctangent function:
  • It is a continuous and monotonic increasing function, meaning it consistently increases as \( x \) becomes larger.
  • As \( x \) approaches infinity, \( \tan^{-1} x \) approaches \( \frac{\pi}{2} \). This limit represents the behavior of an asymptote at \( x = \infty \).
  • At \( x = 0 \), the value of \( \tan^{-1} x \) is 0, which shows its central point of symmetry around the origin.
Understanding these properties helps us predict and analyze the behavior of sequences involving the arctangent function.
Limit of a Sequence
In mathematical analysis, the concept of a sequence limit is fundamental. When a sequence converges, it approaches a specific value called the limit. For the sequence \( a_n = \tan^{-1} n \), we observe its limit as \( n \) approaches infinity. Here's how it typically works:
  • A sequence \( \{a_n\} \) converges to a limit \( L \) if for every small positive number \( \epsilon \), there is a corresponding number \( N \) such that for all \( n > N \), \( |a_n - L| < \epsilon \).
  • In our example, given that \( \tan^{-1} n \) approaches \( \frac{\pi}{2} \) as \( n \) tends towards infinity, it indicates that the sequence \( \{a_n\} \) converges to \( \frac{\pi}{2} \).
Establishing the limit is a critical aspect of confirming convergence in sequences.
Mathematical Analysis
Mathematical analysis involves the study of limits, continuity, and convergence. These are key mechanisms essential for advanced mathematical reasoning. In our context with the sequence \( a_n = \tan^{-1} n \), these concepts illuminate the behavior and convergence details:
  • Analysis helps us break down any function's behavior into manageable pieces, such as determining end behavior and asymptotic tendencies.
  • The precision of analysis ensures that the convergence of \( a_n \) to \( \frac{\pi}{2} \) is validated by rigorous limits and boundedness principles.
  • It provides the tools to describe how sequences approach their limits, making it crucial for verifying assumptions and results accurately.
Harnessing these learning tools enables deeper insight into the long-term behavior of sequences and functions.

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

Does the series $$ \sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{1}{n^{2}}\right) $$ converge or diverge? Justify your answer.

If \(\Sigma a_{n}\) converges and \(a_{n} > 0\) for all \(n\), can anything be said about \(\Sigma\left(1 / a_{n}\right) ?\) Give reasons for your answer.

Prove the "zipper theorem" for sequences: If \(\left\\{a_{n}\right\\}\) and \(\left\\{b_{n}\right\\}\) both converge to \(L,\) then the sequence $$a_{1}, b_{1}, a_{2}, b_{2}, \dots, a_{n}, b_{n}, \dots$$ converges to \(L\).

Assume that each sequence converges and find its limit. \(\sqrt{1}, \sqrt{1+\sqrt{1}}, \sqrt{1+\sqrt{1+\sqrt{1}}}\) \(\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1}}}} \ldots\)

Which of the sequences converge, and which diverge? Give reasons for your answers. \(a_{n}=1-\frac{1}{n}\)
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

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.