/*! 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 44 For the following sequences, plo... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 44

For the following sequences, plot the first 25 terms of the sequence and state whether the graphical evidence suggests that the sequence converges or diverges. \([\mathbf{T}] a_{n}=\sin n\)

Short Answer

Expert verified
The sequence \(a_n = \sin n\) diverges as it fluctuates without convergence.

Step by step solution

01

Understanding the Sequence

The sequence given is \(a_n = \sin n\), where \(n\) is a positive integer. For each natural number \(n\), we take the sine of \(n\), which outputs a real number within the interval [-1, 1].
02

Calculate the First 25 Values

Calculate the sine of each integer from \(n = 1\) to \(n = 25\). For example, \(a_1 = \sin 1\), \(a_2 = \sin 2\), and so on until \(a_{25} = \sin 25\).
03

Plot the Sequence

Plot these 25 values on a graph, with the term number \(n\) on the x-axis and \(a_n\) on the y-axis. This will help visualize how the sequence behaves over these terms.
04

Analyze the Plot

Examine the graph to see if the sequence appears to settle towards a particular value as \(n\) increases or if it continues to fluctuate without settling down.
05

Determine Convergence or Divergence

Since \(\sin n\) does not become closer to a single value as \(n\) increases and instead fluctuates between -1 and 1 without repeating periodically for integers, it suggests that the sequence does not converge. Therefore, the graphical evidence points to divergence.

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.

Sine Function
The sine function, denoted as \( ext{sin}(x)\), is a crucial concept in trigonometry. It takes an angle \(x\) as input and returns its sine, a value that lies within the interval
  • [-1, 1]. The sine of an angle corresponds to the y-coordinate of a unit circle.
  • This means when you graph the sine function, you get a smooth waveform that oscillates between its maximum, 1, and its minimum, -1.
In the context of sequences, the sine function is used to calculate terms based on the position of each term. For instance, \(a_n = \sin n\) computes the sine of each consecutive integer, generating values in the familiar oscillating pattern. This pattern doesn't settle at a single number, which affects whether the sequence converges or diverges.
Graphical Analysis
Graphical analysis involves plotting terms of a sequence to observe its behavior visually. This is particularly helpful in detecting convergence or divergence.
  • For the sequence \(a_n = \sin n\), plotting the values for \(n = 1\) to \(n = 25\) allows us to see if the sequence comes nearer to a particular point.
  • In this case, each point on the graph corresponds to the sine of its respective integer \(n\).
By doing so, you notice that the points do not cluster around any single value. Instead, they scatter between -1 and 1. This widespread distribution is a key indicator of how the sequence behaves, pointing notably towards divergence.
Sequence Behavior
Sequence behavior refers to how the terms of a sequence change as you progress through them. This behavior can be convergent, divergent, or oscillatory.
  • A convergent sequence's terms get closer to a specific value.
  • Divergence, on the other hand, means the terms do not approach a particular value.
  • The sine sequence, \(a_n = \sin n\), demonstrates oscillatory behavior.
Despite being bounded within [-1, 1], it fluctuates across this range without displaying a tendency towards settling at any point. This non-repetitive and non-settling pattern manifests the sequence's divergent nature.
Mathematical Sequences
Mathematical sequences are ordered lists of numbers following a specific rule. Each number in the sequence is called a term.
  • Sequences can be defined by formulas (as in \(a_n = \sin n\)) or by relationships between consecutive terms.
  • Analyzing a sequence often involves exploring whether it converges, diverges, or shows another distinct pattern.
Convergence occurs when terms approach a single finite value as the sequence progresses. The sequence \(a_n = \sin n\) challenges this by never stabilizing to one point. Understanding these fundamental concepts of sequences helps in various mathematical analyses, indicating through \(a_n = \sin n\) that not all sequences converge, underscoring the diversity in mathematical structures.

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

Suppose that \(\sum a_{n}\) converges absolutely. Show that the series consisting of the positive terms \(a_{n}\) also converges.

Let \(0

Use the ratio test to determine whether \(\sum_{n=1}^{\infty} a_{n}\) converges, where \(a_{n}\) is given in the following problems. State if the ratio test is inconclusive. $$\sum_{n=1}^{\infty} \frac{(2 n) !}{n^{2 n}}$$

State whether each of the following series converges absolutely, conditionally, or not at all. $$\sum_{n=1}^{\infty}(-1)^{n+1} 2^{1 / n}$$

[T] Estimates of \(\sum_{n=1}^{\infty} \frac{1}{n^{2}}\) can be accelerated by its \(\quad\) partial \(\quad\) sums writing \(\quad\) its \(\quad\) partial \(\sum_{n=1}^{N} \frac{1}{n^{2}}=\sum_{n=1}^{N} \frac{1}{n(n+1)}+\sum_{n=1}^{N} \frac{1}{n^{2}(n+1)}\) and recalling that \(\sum_{n=1}^{N} \frac{1}{n(n+1)}=1-\frac{1}{N+1}\) converges to one as \(N \rightarrow \infty\). Compare the estimate of \(\pi^{2} / 6\) using the sums \(\sum_{n=1}^{1000} \frac{1}{n^{2}}\) with the estimate using \(1+\sum_{n=1}^{1000} \frac{1}{n^{2}(n+1)} .\)
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Calculus

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.