Problem 4 - Essential Calculus Early Transcendentals

Question

List the first nine terms of the sequence \(\\{\cos (n \pi / 3)\\} .\) Does this sequence appear to have a limit? If so, find it. If not, explain why.

Step-by-Step Solution

Answer

The sequence is periodic and does not have a limit.

Step 1: Understand the Sequence

We are tasked with finding the first nine terms of the sequence given by \( \{\cos(n \pi / 3)\} \). Here, the expression for each term of the sequence is \( \cos(n \pi / 3) \), where \( n \) is a positive integer. We will substitute \( n \) starting from 1 up to 9.

Step 2: Calculate the First Term

Substitute \( n = 1 \) into the sequence formula: \[ \cos(1 \pi / 3) = \cos(\pi / 3) = \frac{1}{2}. \] Therefore, the first term of the sequence is \( \frac{1}{2} \).

Step 3: Calculate the Second Term

Substitute \( n = 2 \): \[ \cos(2 \pi / 3) = \cos(\pi - \pi / 3) = -\cos(\pi / 3) = -\frac{1}{2}. \] The second term is \( -\frac{1}{2} \).

Step 4: Calculate the Third Term

Substitute \( n = 3 \): \[ \cos(3 \pi / 3) = \cos(\pi) = -1. \] Thus, the third term is \(-1\).

Step 5: Calculate the Fourth Term

Substitute \( n = 4 \): \[ \cos(4 \pi / 3) = \cos(\pi + \pi / 3) = -\cos(\pi / 3) = -\frac{1}{2}. \] The fourth term is \(-\frac{1}{2}\).

Step 6: Calculate the Fifth Term

Substitute \( n = 5 \): \[ \cos(5 \pi / 3) = \cos(2\pi - \pi / 3) = \cos(\pi / 3) = \frac{1}{2}. \] The fifth term is \( \frac{1}{2} \).

Step 7: Calculate the Sixth Term

Substitute \( n = 6 \): \[ \cos(6 \pi / 3) = \cos(2\pi) = 1. \] Hence, the sixth term is \(1\).

Step 8: Calculate the Seventh Term

Substitute \( n = 7 \): \[ \cos(7 \pi / 3) = \cos(2\pi + \pi / 3) = \cos(\pi / 3) = \frac{1}{2}. \] The seventh term is \( \frac{1}{2} \).

Step 9: Calculate the Eighth Term

Substitute \( n = 8 \): \[ \cos(8 \pi / 3) = \cos(3\pi - \pi / 3) = -\cos(\pi / 3) = -\frac{1}{2}. \] The eighth term is \(-\frac{1}{2}\).

Step 10: Calculate the Ninth Term

Substitute \( n = 9 \): \[ \cos(9 \pi / 3) = \cos(3\pi) = -1. \] The ninth term is \(-1\).

Step 11: Analyze for a Limit

The first nine terms of the sequence are: \( \frac{1}{2}, -\frac{1}{2}, -1, -\frac{1}{2}, \frac{1}{2}, 1, \frac{1}{2}, -\frac{1}{2}, -1 \). These terms repeat in a cycle of \( \frac{1}{2}, -\frac{1}{2}, -1, -\frac{1}{2}, \frac{1}{2}, 1 \), showing a non-converging periodic behavior. Thus, the sequence does not have a limit because it does not converge to a single value.

Key Concepts

Trigonometric Sequences

Trigonometric sequences are series that involve trigonometric functions like sine, cosine, or tangent. In our exercise, we deal with the cosine function within the sequence \(\{\cos(n \pi / 3)\}\). Trigonometric sequences can oscillate due to the cyclical nature of trigonometric functions.

If you remember the unit circle, the cosine of angles is periodic, meaning that it repeats every full circle, or every 360 degrees (which is \(2\pi\) radians).

• For instance, \(\cos(0) = \cos(2\pi) = \cos(4\pi) = 1\).
• This pattern helps us understand the terms of our sequence, as each \(\pi / 3\) increment moves to another part of the circle.

The trigonometric sequence \(\{\cos(n \pi / 3)\}\) displays values such as \(\frac{1}{2}, -\frac{1}{2}, -1\), among others, relying on the angle within the circle.

Limits of Sequences

The concept of a limit of a sequence helps us determine if a sequence settles into a single value as its terms progress to infinity. However, for some sequences, they do not settle into just one specific number, meaning they do not converge.

When analyzing our trigonometric sequence, you might instinctively look for a pattern or a limit as the terms go on. But the repeating cycle tells us otherwise.

• We see terms like \(\frac{1}{2}, -\frac{1}{2}, -1, \text{and} 1\) repeating rather than approaching a particular value.
• The fluctuation amongst these terms means that the sequence does not have a limit.

The purpose of identifying limits is often to understand stability in behavior as the sequence "grows." In cases like ours, periodic sequences will generally resist convergence due to their shifting nature.

Periodic Sequences

Periodic sequences consist of terms that repeat after a certain number of steps. These sequences manifest a clear cycle or repetition pattern in their elements.

In our sequence \(\{\cos(n \pi / 3)\}\), after a series of six terms, the pattern clearly begins repeating. The terms of the cycle \(\frac{1}{2}, -\frac{1}{2}, -1, -\frac{1}{2}, \frac{1}{2}, 1\) cycle back, illustrating the repetitive nature of trigonometric functions on the unit circle.

• Periodic sequences do not tend to have a limit because they do not stabilize at a particular point or value.
• This periodic nature is crucial in the understanding of various mathematical and real-world phenomena, such as waves and oscillations.

So, in the context of the exercise, we see that the sequence does not converge but rather repeats, which is the hallmark of periodic sequences.