Problem 65 - Precalculus with Limits

Question

In Exercises 65-68, create a scatter plot of the terms of the sequence. Determine whether the sequence converges or diverges. If it converges, estimate its limit. $$a_n = 4\left(\dfrac{2}{3} \right)^n $$

Step-by-Step Solution

Answer

The sequence \(a_n = 4\left(\dfrac{2}{3}\right)^n \) both converges and its limit as \(n\) approaches infinity is 0.

Step 1: Plotting the Sequence

The sequence is \(a_n = 4\left(\dfrac{2}{3}\right)^n \). To plot this, create a scatter plot where the x-axis represents the term number (n) and the y-axis represents the term value ( \(a_n\) ). For instance, when \(n=1\), \(a_1 = 4\left(\dfrac{2}{3}\right)^1 = \dfrac{8}{3}\). Continue this for a range of n values to generate the scatter plot.

Step 2: Determine Convergence or Divergence

A sequence is said to converge if, as \(n\) goes to infinity, the sequence approaches a specific value. A sequence diverges if it does not approach a specific value. In the case of this question, as \(n\) goes to infinity, \( \left(\dfrac{2}{3}\right)^n \) goes to zero because it's a fraction with denominator greater than the numerator. Multiplied by 4, the sequence still equals zero. Therefore, the sequence vanishes, which implies convergence.

Step 3: Estimate the Limit

The limit of a sequence is the value the sequence converges to as \(n\) approaches infinity. Here, as \(n\) goes to infinity, \(a_n = 4\left(\dfrac{2}{3}\right)^n \) goes to zero. Thus, the limit of the sequence is 0.

Key Concepts

Scatter Plot

A scatter plot is a useful visual tool that helps in understanding the behavior of a sequence or data points. In this case, it allows us to observe how the values of the terms in the sequence change as the index \(n\) increases.

• The x-axis represents the index \(n\), which is essentially the position of the term in the sequence.
• The y-axis shows the value of the sequence term \(a_n\).

To create an accurate scatter plot for the given sequence \(a_n = 4\left(\dfrac{2}{3}\right)^n\), plot multiple points by calculating \(a_n\) for several consecutive \(n\) values. This approach clearly illustrates the trend or pattern of the sequence.
This visualization makes it easier to assess patterns such as convergence or divergence, as well as the general rate at which the values approach a limit.

Limit of a Sequence

The concept of a limit is central when studying sequences. The limit of a sequence is the value that the terms in the sequence get closer to as the index \(n\) becomes very large.

• If the terms approach a specific number, then we say the sequence converges to that limit.
• If they do not approach a specific number, we say it diverges.

For the sequence \(a_n = 4\left(\dfrac{2}{3}\right)^n\), as \(n\) increases, the term \(\left(\dfrac{2}{3}\right)^n\) becomes smaller and approaches zero because it's a fraction less than 1 being raised to successively higher powers.
As a result, the whole sequence also approaches zero because multiplying it by 4 does not change the rate it approaches zero, thus establishing the limit of the sequence as 0. Understanding limits is crucial for interpreting the behavior of sequences in mathematics.

Convergence and Divergence

Convergence and divergence describe how sequences behave as the index \(n\) grows.

• A sequence is said to converge if its terms approach a certain finite number—a limit.
• If a sequence's terms do not approach any specific value, we say it diverges.

In the given sequence \(a_n = 4\left(\dfrac{2}{3}\right)^n\), it converges because as \(n\) reaches higher values, \(\left(\dfrac{2}{3}\right)^n\) approaches zero. This fraction gets smaller and approaches zero since the numerator is smaller than the denominator.
This behavior indicates that no matter how large the term number \(n\) becomes, the value of \(a_n\) gets closer and closer to zero. That's a prime example of sequence convergence. Contrarily, had \(a_n\) been such that it grows indefinitely in magnitude as \(n\) increases, it would have diverged.