Problem 26 - Calculus: Early Transcendentals

Question

Determine whether the sequence converges or diverges. If it converges, find the limit. \( a_n = 2 + (0.86)^n \)

Step-by-Step Solution

Answer

The sequence converges, and its limit is 2.

Step 1: Identify the Sequence

First, let's identify the sequence given in the question: The sequence is: \( a_n = 2 + (0.86)^n \).

Step 2: Analyze the Behavior of the Exponential Term

Observe the term \( (0.86)^n \). As \( n \) becomes larger, the term \( (0.86)^n \) approaches 0 because 0.86 is between 0 and 1, meaning that multiplying it by itself repeatedly will make it shrink.

Step 3: Determine the Convergence of the Sequence

Since \( (0.86)^n \) approaches 0 as \( n \rightarrow \infty \), then the behavior of \( a_n \) will be dominated by the constant term, which is 2. Thus, the sequence \( a_n = 2 + (0.86)^n \) converges.

Step 4: Find the Limit of the Sequence

The limit of \( a_n = 2 + (0.86)^n \) as \( n \rightarrow \infty \) is:\[ \lim_{{n \to \infty}} a_n = \lim_{{n \to \infty}} (2 + (0.86)^n) = 2 + 0 = 2. \] Therefore, the limit of the sequence is 2.

Key Concepts

Limit of a Sequence

The limit of a sequence is a fundamental concept in mathematics, especially when studying sequences and their behavior as they extend towards infinity. Understanding the limit of a sequence helps determine where the sequence stabilizes or heads towards when its terms grow larger and larger.If a sequence approaches a specific value as the number of terms increases, we say it "converges" to that limit. A practical way to visualize this is by imagining the terms of the sequence gradually focusing on a particular value, getting closer with each step.For example, consider the sequence:

• The sequence given is: \( a_n = 2 + (0.86)^n \).
• As the term \((0.86)^n\) approaches zero, the sequence values approach the constant 2.
• Eventually, however large \(n\) becomes, \( a_n \) will come very close to 2 but never quite reach it - this leads to the conclusion: \ \(\lim_{{n \to \infty}} a_n = 2\).

Understanding limits is crucial because it allows us to predict the long-term behavior of sequences across various mathematical and real-world applications.

Exponential Decay

Exponential decay is a process where quantities diminish rapidly at a rate proportional to their current value. In mathematical terms, when a sequence or function decreases multiplicatively by a constant factor every step, it exhibits exponential decay. This concept is vital in fields ranging from biology to finance, where rapid changes occur over time. Consider the sequence:

• The term \((0.86)^n\) demonstrates exponential decay.
• Since 0.86 is less than 1, raising it to consecutive powers (i.e., multiplying it by itself over and over) results in smaller values.
• The term \((0.86)^n\) approaches zero as \(n\) increases, which crucially impacts the sequence \( a_n = 2 + (0.86)^n \).

Exponential decay is a useful concept for understanding how different processes evolve over time, particularly when projecting long-term outcomes.

Convergence and Divergence

The concepts of convergence and divergence refer to the behavior of sequences or series as their index becomes very large. Knowing whether a sequence converges or diverges is essential for analyzing its stability and predicting future terms.

• A sequence is said to "converge" if its terms approach a single value, known as the limit.
• If the terms of a sequence do not settle down to a definite value, the sequence is said to "diverge."
• For the sequence \( a_n = 2 + (0.86)^n \), - As \((0.86)^n\) decreases towards zero, the sequence converges to the limit 2.

Recognizing whether sequences converge or diverge can help predict their long-term behavior, giving insight into the system or phenomenon being studied.