Problem 27 - Calculus: Early Transcendentals

Question

Determine whether the sequence converges or diverges. If it converges, find the limit. \( a_n = 3^n 7^{-n} \)

Step-by-Step Solution

Answer

The sequence converges, and its limit is 0.

Step 1: Express the Sequence

We start with the given sequence: \( a_n = 3^n 7^{-n} \). This can be rewritten by combining the exponents: \( a_n = \left( \frac{3}{7} \right)^n \).

Step 2: Identify the Base of the Exponential

The sequence \( a_n = \left( \frac{3}{7} \right)^n \) is an exponential sequence with a base of \( \frac{3}{7} \). This base is less than 1, which is key for evaluating convergence.

Step 3: Determine Convergence

Since the base \( \frac{3}{7} \) is between 0 and 1, the sequence \( a_n = \left( \frac{3}{7} \right)^n \) converges to 0 as \( n \to \infty \). This is a known property of exponential sequences with bases in this range.

Step 4: State the Limit

Therefore, the limit of the sequence as \( n \to \infty \) is 0, which confirms that the sequence converges to 0.

Key Concepts

Exponential Sequences

An exponential sequence is a type of sequence where each term is a constant raised to the power of its position in the sequence. In mathematical terms, this is described by the expression \( a_n = a^n \), where \( a \) is a constant. For our exercise example, the initiated sequence \( a_n = 3^n 7^{-n} \) can be simplified by recognizing it as \( a_n = \left( \frac{3}{7} \right)^n \) once the exponents are combined. This indicates that the sequence is exponential with a base \( \frac{3}{7} \).

Exponential sequences are widely recognized for their predictable behavior:

• If the base \( |a| > 1 \), the sequence tends to diverge, meaning it will grow larger and larger without approaching any particular value.
• If the base \( |a| < 1 \), the sequence converges toward zero. This occurs because exponential growth becomes exponential decay when the base is a fraction less than one.

In our case, with a base of \( \frac{3}{7} \), which is less than one, the exponential sequence will approach zero.

Convergence Criteria

To determine if a sequence converges, you need to evaluate the ultimate behavior of its terms as they approach infinity. For exponential sequences, convergence often hinges on the value of the base:

• If the base is greater than one, the sequence will diverge.
• When the base is less than one, you can expect the sequence to shrink towards zero, which indicates convergence.

For the sequence \( a_n = \left( \frac{3}{7} \right)^n \), we noticed that the base \( \frac{3}{7} \) is a fraction between 0 and 1. This is a determiner of convergence because fractions in this interval repeatedly raised by consecutive integers (n) slice the value smaller and smaller, drawing nearer to zero.

Another aspect of convergence criteria is the ability to ascertain a limit, which acts as a confirmation of the sequence's destination in terms of value as \( n \to \infty \). Keep in mind, criteria differ slightly depending on the type of sequence, making it essential to recognize the sequence type first.

Limits of Sequences

The limit of a sequence is a pivotal concept which tells us the value that the sequence tends toward as the number of terms becomes infinitely large. The key idea is to monitor whether a sequence reaches a particular value, no matter how large the term number becomes. Limits help validate convergence.

For our specific sequence \( a_n = \left( \frac{3}{7} \right)^n \):

• It approaches the limit of \( 0 \) as \( n \to \infty \), because repeatedly multiplying fractions less than one results in progressively smaller and smaller values that zero in on zero.
• Practically speaking, if after numerous iterations the terms of your sequence remain extremely close to a single value or within a permissible margin (often acknowledged as an adequate proximity), you can assert a limit has been achieved.

Understanding limits allows students to grasp "end behavior” of sequences and predict long-term outcomes, reflecting a sequence's potential to stabilize or perpetually grow.