Problem 72 - Calculus: Early Transcendentals

Question

Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? \( a_n = \cos n \)

Step-by-Step Solution

Answer

The sequence is not monotonic and is bounded by -1 and 1.

Step 1: Understand the Sequence

The given sequence is \( a_n = \cos n \). This means that each term of the sequence is the cosine of an integer angle \( n \). The sequence alternates as \( n \) changes, due to the periodic nature of the cosine function.

Step 2: Determine if the Sequence is Monotonic

Recall that a sequence is increasing if each term is greater than or equal to the previous term and decreasing if each term is less than or equal to the previous term. Since \( \cos n \) does not consistently increase or decrease as \( n \) increases due to the oscillating nature of the cosine function, the sequence is not monotonic.

Step 3: Check if the Sequence is Bounded

A sequence is bounded if there is a real number such that all terms of the sequence are less than or equal to it and greater than or equal to another real number. Since \( \cos n \) always lies between -1 and 1, the sequence \( a_n = \cos n \) is bounded by these values.

Key Concepts

Monotonic Sequences

To determine if a sequence is monotonic, we need to check if it consistently increases or decreases. A sequence is said to be increasing if each term is greater than or equal to the previous term. Conversely, it's decreasing if each term is less than or equal to the prior term.
Monotonic sequences have a predictable behavior where no term violates the increasing or decreasing order.
In contrast, non-monotonic sequences do not have any such consistent trend. For instance, the sequence defined by \( a_n = \cos n \) doesn't follow a steady increase or decrease pattern.
This is because the cosine function oscillates as \( n \) takes integer values. It fluctuates between -1 and 1, thus defying any monotonic behavior.

Bounded Sequences

A sequence is bounded if it stays within a set range for all its terms. Specifically, a sequence is upper-bounded if there is a maximum value that no term exceeds and lower-bounded if there is a minimum that no term goes below.
For the sequence \( a_n = \cos n \), the terms are always contained within the interval [-1, 1]. This is due to the intrinsic properties of the cosine function, which never exceeds these bounds.

• Upper bound: 1
• Lower bound: -1

Therefore, \( a_n = \cos n \) is a bounded sequence, making it both upper and lower bounded within this specified range.

Cosine Function

The cosine function, denoted as \( \cos \), is a fundamental trigonometric function. It is periodic, with a period of \( 2\pi \), meaning that it repeats its values every \( 2\pi \) radians.
This periodic nature contributes to its oscillating behavior, especially evident when cosine is evaluated at integer values of \( n \).

• The range of \( \cos(x) \) is \([-1, 1]\), which determines the bounds of the related sequence.
• The graph of \( \cos \,x \) is wave-like, illustrating its rise and fall between its maximum and minimum values.

In terms of sequences, such as \( a_n = \cos n \), these characteristics dictate how the sequence behaves over time.

Oscillating Sequences

An oscillating sequence is one that does not settle into a monotonic pattern but instead fluctuates. These sequences swing back and forth between values, without converging towards a specific number.
The sequence \( a_n = \cos n \) is a classic example of an oscillating sequence due to the nature of the cosine function.

• It cycles through values repeatedly.
• It does not approach a fixed limit as \( n \) increases.

Oscillating sequences are interesting since they demonstrate periodic behavior without exhibiting stability or convergence associated with steady sequences.