Problem 29 - Precalculus: Mathematics for Calculus

Question

Limits of Sequences If the sequence with the given \(n\) th term is convergent, find its limit. If it is divergent, explain why. $$a_{n}=\sin (n \pi / 2)$$

Step-by-Step Solution

Answer

The sequence is divergent because it cycles through 1, 0, -1, 0 and does not approach a single value.

Step 1: Evaluate the Sequence at Specific Terms

Let's evaluate several terms of the sequence to understand its pattern. For \( n = 1 \), \( a_1 = \sin(\pi/2) = 1 \). For \( n = 2 \), \( a_2 = \sin(\pi) = 0 \). For \( n = 3 \), \( a_3 = \sin(3\pi/2) = -1 \). For \( n = 4 \), \( a_4 = \sin(2\pi) = 0 \). Clearly, the sequence alternates between 1, 0, -1, and 0.

Step 2: Determine the Sequence Pattern

Observe the evaluated terms from Step 1. The sequence is periodic with a period of 4, repeating the values \(1, 0, -1, 0\).

Step 3: Convergence Analysis

To determine convergence, consider whether the terms approach a single value as \( n \to \infty \). Since the sequence keeps cycling through multiple values (1, 0, -1, 0) and does not approach a specific value, it does not converge.

Step 4: Conclusion on Divergence

Since the sequence does not settle to a single, finite value, it is divergent. The periodic nature prevents convergence.

Key Concepts

Convergent Sequences

A sequence is called **convergent** if its terms approach a specific value as the number of terms progresses to infinity. In simpler terms, if you keep adding terms to a convergent sequence, eventually all the terms will be very close to one specific number.
This specific number is known as the **limit** of the sequence.
For example:

• The sequence given by \(a_n = \frac{1}{n}\) converges to 0. As \(n\) increases, the terms get closer and closer to 0.

Convergent sequences are important because they demonstrate predictability and stability, making them useful in mathematical analysis and real-world applications.

Divergent Sequences

A sequence is defined as **divergent** if it does not approach any specific limit. This means that as you examine more terms in the sequence, the terms don’t settle on any one number.
Let's consider the trigonometric sequence from our exercise: \(a_n = \sin(n \pi / 2)\). As evaluated earlier, this sequence doesn't settle to a particular value and keeps cycling through multiple values (1, 0, -1, 0).

• This lack of settling indicates divergence.
• Divergent sequences do not have a single limit point.

Such sequences can seem chaotic or random, and they illustrate scenarios where stability isn't achieved.

Sequence Periodicity

**Periodicity** in sequences refers to the repeating pattern of terms. If a sequence repeats its values in a regular cycle, it is called periodic.
For example, in our exercise, the sequence \(a_n = \sin(n \pi / 2)\) exhibits a periodic pattern with a period of 4. This means every four terms, the pattern repeats (1, 0, -1, 0).

• Periodic sequences have intervals, or periods, that are consistent.
• The "period" of a sequence is the number of terms before the sequence starts repeating.

Recognizing periodicity is helpful for predicting future terms of a sequence and understanding its behavior over time.

Trigonometric Sequences

**Trigonometric sequences** are sequences whose terms are defined using trigonometric functions, such as sine or cosine. These sequences often showcase interesting behaviors, like periodicity and oscillation.
The sequence from the exercise, \(a_n = \sin(n \pi / 2)\), is a perfect example. It shows how trigonometric functions can create predictable, yet complex patterns.

• Trigonometric sequences can vary in amplitude and cycle depending on the function and its argument (the term input).
• They often repeat their values at regular intervals, which is another indication of periodicity.

Understanding trigonometric sequences requires familiarity with fundamental properties of sine, cosine, and other trig functions, including their specific periodic behaviors and value ranges.