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Chapter 9: Problem 27

Do the sequences, converge or diverge? If a sequence converges, find its limit. $$\cos (\pi n)$$

Short Answer

Expert verified
The sequence diverges as it alternates between -1 and 1.

Step by step solution

01

Understand the Sequence

The given sequence is \( \cos(\pi n) \), where \( n \) is a sequence of natural numbers \( n = 1, 2, 3, \ldots \). We need to determine if this sequence converges to a single value or diverges.
02

Compute the First Few Terms

Calculate the first few terms of the sequence to identify any pattern. For \( n = 1, 2, 3, 4 \), we have: \( \cos(\pi \times 1) = -1 \), \( \cos(\pi \times 2) = 1 \), \( \cos(\pi \times 3) = -1 \), \( \cos(\pi \times 4) = 1 \). The sequence alternates between -1 and 1.
03

Identify the Pattern

From the previous step, we see that the sequence is periodic with period 2: \( -1, 1, -1, 1, \ldots \). This repetitive pattern indicates that the sequence does not approach a single value.
04

Analyze Convergence or Divergence

A sequence converges if the terms eventually get arbitrarily close to a specific number. Since \( \cos(\pi n) \) alternates between -1 and 1, it does not get closer to any single value as \( n \) increases. Hence, the sequence diverges.

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Key Concepts

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

Cosine Function
The cosine function, denoted as \( \cos(x) \), is a fundamental trigonometric function that describes the x-coordinate of a point on the unit circle at a specific angle \( x \). It is defined for all real numbers, and its range lies between -1 and 1. This function is periodic, meaning it repeats its values in a regular cycle. For the cosine function, this cycle occurs every \( 2\pi \) radians, making it useful in analyzing periodic behaviors.
  • The cosine of 0 is 1,
  • The cosine of \( \pi/2 \) is 0,
  • The cosine of \( \pi \) is -1,
  • The cosine of \( 3\pi/2 \) is 0.
Understanding the periodic nature and symmetry of the cosine function helps in solving problems involving trigonometric identities and sequence evaluations.
Divergence
Divergence is a term that describes a sequence or series failing to converge to a limit. For a sequence to be considered divergent, its terms do not settle down to a fixed value as the sequence progresses. In the context of the sequence \( \cos(\pi n) \), as \( n \) increases, the sequence alternates between -1 and 1.
Since there is no single number that the sequence terms approach, the sequence is said to diverge. It does not stabilize on one number but rather repeats in a non-converging manner.
  • Divergent sequences do not have a finite limit.
  • An analysis of divergence often involves checking if a sequence does not settle or approach a meaningful limit.
Divergence is a critical concept in mathematics, particularly in calculus and analysis, helping to categorize sequences and their behaviors.
Periodic Sequences
Periodic sequences are sequences whose terms repeat at regular intervals. They are characterized by a fixed pattern that continues indefinitely. For example, the sequence \( \cos(\pi n) \) is periodic, as it repeats the values -1 and 1 every two terms.Periodic sequences have a fundamental period, which is the smallest number of terms after which the pattern repeats. In our sequence, since \( \cos(\pi \times n) \) alternates between -1 and 1, it has a period of 2.
  • Periodic sequences do not converge to a single value.
  • They are useful for modeling cyclical phenomena in science and engineering.
  • The periodic nature is closely related to trigonometric functions and their properties.
Understanding periodic sequences is crucial when solving problems involving repeating patterns or cycles in mathematical contexts.

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Most popular questions from this chapter

Decide if the statements are true or false. Give an explanation for your answer. If a sequence \(s_{n}\) is convergent, then the terms \(s_{n}\) tend to zero as \(n\) increases.

Determine whether the series converges. $$\sum_{n=1}^{\infty} \frac{\sin n}{n^{2}}$$

Explain what is wrong with the statement. The sum of the infinite geometric series \(1-\frac{3}{2}+\frac{9}{4}-\) \(\frac{27}{8}+\dots\) is \(\frac{1}{1+3 / 2}=\frac{2}{5}\)

If \(\sum\left|a_{n}\right|\) converges, then \(\sum(-1)^{n}\left|a_{n}\right|\) converges.

Explain what is wrong with the statement. The series \(\sum_{n=1}^{\infty} 1 /\left(n^{2}+1\right)\) converges by the ratio test.
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