Problem 14 - Essential Calculus Early Transcendentals

Question

\(9-32\) n Determine whether the sequence converges or diverges. If it converges, find the limit. $$a_{n}=\frac{3^{n+2}}{5^{n}}$$

Step-by-Step Solution

Answer

The sequence converges, and the limit is 0.

Step 1: Understand the Sequence

The given sequence is \( a_{n} = \frac{3^{n+2}}{5^{n}} \). We need to analyze this sequence to determine whether it converges or diverges as \( n \to \infty \).

Step 2: Simplify the Expression

Simplify the expression \( a_{n} = \frac{3^{n+2}}{5^{n}} \) by rewriting the numerator: \( 3^{n+2} = 3^n \cdot 3^2 = 9 \cdot 3^n \). Thus, the sequence becomes \( a_{n} = \frac{9 \cdot 3^n}{5^n} \).

Step 3: Express in Terms of a Single Base

Rewrite the expression \( a_{n} = \frac{9 \cdot 3^n}{5^n} \) as \( 9 \cdot \left(\frac{3}{5}\right)^n \). This shows that the sequence can be expressed as a geometric sequence with the base \( \frac{3}{5} \).

Step 4: Determine Ratio's Behavior as n Increases

Since \( \left(\frac{3}{5}\right)^n \) has a base \( \frac{3}{5} < 1 \), we know that as \( n \to \infty \), \( \left(\frac{3}{5}\right)^n \to 0 \).

Step 5: Conclusion on Convergence and Limit

Multiply the limit of \( \left(\frac{3}{5}\right)^n \to 0 \) by the constant 9: \( 9 \cdot 0 = 0 \). Thus, the sequence \( a_{n} \) converges, and its limit is 0.

Key Concepts

Geometric Sequences

Geometric sequences are a pattern of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
For example, in the sequence \( 2, 6, 18, 54, \dots \), each term is multiplied by 3, which is the common ratio.

• For a sequence to qualify as geometric, each term must be obtained by multiplying the previous one by the same number.
• The general form of a geometric sequence can be expressed as \( a, ar, ar^2, ar^3, \ldots \).
• Here, \( a \) is the first term, and \( r \) is the common ratio.

The sequence from the exercise, \( a_n = 9 \left( \frac{3}{5} \right)^n \) exemplifies a geometric sequence.
The fixed number, \( \frac{3}{5} \), serves as the common ratio here.

Limit of Sequences

The limit of a sequence is the value that the terms of a sequence "approach" as the index (often n) goes to infinity.
Finding the limit of sequences helps us understand the long-term behavior.

• If a sequence has a limit, it is called convergent. Otherwise, it is divergent.
• For our example, the sequence is \( a_n = 9 \left( \frac{3}{5} \right)^n \). The common ratio \( \frac{3}{5} \) is less than 1.
• When the common ratio is less than 1, the terms of the sequence tend to zero as \( n \to \infty \).

Thus, the sequence simply heads towards zero, indicating convergence.

Mathematical Analysis

Mathematical analysis provides the tools for rigorously studying the behavior and properties of sequences.

• Analysis often involves considering limits, continuity, and convergence.
• When analyzing sequences, we seek to understand not just the pattern, but where it 'settles' as the terms increase indefinitely.
• For the sequence \( a_n = \frac{3^{n+2}}{5^n} \), simplifying to \( 9 \left( \frac{3}{5} \right)^n \) helps us employ analysis to conclude that it converges to 0.
• This showcases how analysis can simplify more complicated expressions to reveal their true nature.

Through mathematical analysis, we can confidently determine the behavior of sequences over infinite horizons.