Problem 27 - 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. $$a_{n}=\frac{1}{3^{n}}$$

Step-by-Step Solution

Answer

The sequence converges to the limit 0.

Step 1: Determine if the sequence is convergent or divergent

The sequence given is \(a_n = \frac{1}{3^n}\). To determine if it's convergent, we need to analyze the behavior of the sequence as \(n\) approaches infinity. As \(n\) becomes very large, \(3^n\) increases exponentially while the numerator remains constant at 1. This causes \(a_n\) to approach 0. Therefore, the sequence is convergent.

Step 2: Find the limit of the sequence

Since we determined that the sequence converges, we need to calculate its limit. As \(n\rightarrow \infty\), we see that \(\frac{1}{3^n}\) approaches 0 because the exponentiation makes \(3^n\) significantly larger, pushing the fraction closer to zero. Hence, \(\lim_{{n\to\infty}} \frac{1}{3^n} = 0\).

Key Concepts

Convergent Sequence

A convergent sequence is one that approaches a specific value as its index, usually denoted by \( n \), goes to infinity. In simpler terms, as you keep extending the sequence, the terms get closer and closer to a fixed number. Take the sequence \( a_n = \frac{1}{3^n} \) as an example. As \( n \) becomes very large, the denominator \( 3^n \) grows at a much faster rate than the numerator, which remains constant at 1.
This rapid increase in the term's denominator makes the fraction smaller and closer to zero. Hence, the sequence \( a_n = \frac{1}{3^n} \) converges to 0.
Properties of convergent sequences include:

• They approach a specific limit as \( n \rightarrow \infty \).
• The terms of the sequence get arbitrarily close to this limit.
• Convergent sequences contain patterns or predictability.

This understanding helps in various fields, like calculus, where knowing the behavior of a sequence aids in analyzing functions.

Divergent Sequence

Unlike convergent sequences, divergent sequences do not approach a specific limit as \( n \) increases. They might increase indefinitely, decrease indefinitely, or exhibit erratic fluctuations without settling around a stable value. Divergent sequences are important to understand for recognizing the limits and behavior of functions and sequences.
For instance, consider a sequence defined by \( a_n = n \). As \( n \rightarrow \infty \), \( a_n \) continues to increase without bounds, indicating the sequence is divergent.
Key characteristics of divergent sequences include:

• Lack of a specific limit or value as \( n \rightarrow \infty \).
• Potential for unbounded growth or decline.
• Absence of stability, often used to demonstrate scenarios beyond normal bounds.

Understanding the nature of divergence helps in identifying whether certain mathematical functions can be evaluated at specific points or how they behave at infinity.

Exponential Growth

Exponential growth refers to an ever-increasing rate of growth, meaning the quantity becomes significantly larger as time goes on. In the sequence provided, \( a_n = \frac{1}{3^n} \), the exponential growth is seen in the denominator \( 3^n \), which increases very quickly as \( n \) increases. This rapid growth of the denominator is what leads the sequence to converge to zero.
Exponential growth is characterized by:

• A base raised to an increasing power \( n \), leading to rapid increase or decrease depending on context.
• A constant percentage rate of growth, typical for certain populations or investments.
• Often contrasts with linear growth, which increases by additive constants rather than multiplicative factors.

This concept is crucial in natural sciences, finance, and technology, where understanding the pace and impact of growth is essential for predictions and planning.