Problem 22 - Calculus: Early Transcendentals

Question

Calculate, to four decimal places, the first ten terms of the sequence and use them to plot the graph of the sequence by hand. Does the sequence appear to have a limit? If so, calculate it. If not, explain why. \( a_n = 1 + \frac{10^n}{9^n} \)

Step-by-Step Solution

Answer

The sequence does not have a limit as it diverges to infinity.

Step 1: Identify the Sequence Formula

The sequence given is \( a_n = 1 + \frac{10^n}{9^n} \). This will be used to calculate each term.

Step 2: Calculate the First Ten Terms

We will calculate the first ten terms using the formula \( a_n = 1 + \frac{10^n}{9^n} \), evaluating \( n = 1 \) to \( n = 10 \).- For \( n = 1 \), \( a_1 = 1 + \frac{10^1}{9^1} = 1 + \frac{10}{9} \approx 2.1111 \).- For \( n = 2 \), \( a_2 = 1 + \frac{10^2}{9^2} = 1 + \frac{100}{81} \approx 2.2346 \).- For \( n = 3 \), \( a_3 = 1 + \frac{10^3}{9^3} = 1 + \frac{1000}{729} \approx 2.3718 \).- For \( n = 4 \), \( a_4 = 1 + \frac{10^4}{9^4} = 1 + \frac{10000}{6561} \approx 2.5235 \).- For \( n = 5 \), \( a_5 = 1 + \frac{10^5}{9^5} = 1 + \frac{100000}{59049} \approx 2.6927 \).- For \( n = 6 \), \( a_6 = 1 + \frac{10^6}{9^6} = 1 + \frac{1000000}{531441} \approx 2.8821 \).- For \( n = 7 \), \( a_7 = 1 + \frac{10^7}{9^7} = 1 + \frac{10000000}{4782969} \approx 3.0941 \).- For \( n = 8 \), \( a_8 = 1 + \frac{10^8}{9^8} = 1 + \frac{100000000}{43046721} \approx 3.3317 \).- For \( n = 9 \), \( a_9 = 1 + \frac{10^9}{9^9} = 1 + \frac{1000000000}{387420489} \approx 3.5987 \).- For \( n = 10 \), \( a_{10} = 1 + \frac{10^{10}}{9^{10}} = 1 + \frac{10000000000}{3486784401} \approx 3.8990 \).

Step 3: Plot the Sequence Graph

Using the values calculated for \( a_1 \) to \( a_{10} \), plot the points on a graph with \( n \) as the x-axis and \( a_n \) as the y-axis. This will help visualize the behavior of the sequence.

Step 4: Determine if the Sequence Has a Limit

Observe the graph and the calculated terms. Note that as \( n \) increases, the terms \( a_n \) appear to grow without bound, suggesting that the sequence does not converge to a finite limit. The sequence diverges as \( n \) approaches infinity, because the ratio \( \left(\frac{10}{9}\right)^n \) grows larger than any fixed number.

Key Concepts

Limit of a Sequence

In mathematics, the limit of a sequence is a fundamental concept. It describes the value that the terms of a sequence "approach" as the index, usually denoted by \( n \), goes to infinity. Simply put, if the terms of a sequence get closer and closer to a specific number as \( n \) increases, we say that the sequence converges to that limit.
For a sequence to have a limit:

• The difference between the terms and the limit must become arbitrarily small as you go further along in the sequence.
• The limit should be a finite number.

Let's look at our sequence \( a_n = 1 + \frac{10^n}{9^n} \). As we calculated the first ten terms, we observed these numbers grow. This pattern suggests that the terms are not approaching a particular fixed number, hence the sequence may not have a finite limit.
Intuitively, if a sequence doesn't stabilize at a certain value, it can't have a limit. It's important to recognize this is a critical signal of non-converging behavior.

Divergence

A sequence is said to be divergent if it doesn't converge to a limit. When a sequence grows or decreases without any bounds, it diverges. Divergence is like a runaway train in math—it keeps going.

• If the terms of a sequence consistently grow larger or smaller without settling at any particular value, you can classify it as divergent.
• In our exercise's sequence \( a_n = 1 + \frac{10^n}{9^n} \), we see that the terms become significantly larger as \( n \) increases, strengthening the claim of divergence.

This lack of settlement observed indicates that as \( n \rightarrow \infty \), the sequence \( a_n \) doesn't approach a constant. Additionally, looking at the term \( \left(\frac{10}{9}\right)^n \), it becomes larger than any fixed number as \( n \) grows. Thus, the sequence diverges.

Calculus Graphing

Graphing is a fantastic tool in calculus to visualize sequences and understand their behavior over time. It can reveal characteristics like convergence or divergence more vividly by providing a visual cue.
When plotting a sequence, you'll typically:

• Use the sequence values (\( a_n \)) for the y-axis.
• Use the index values (\( n \)) for the x-axis.

A graph of our provided sequence, \( a_n = 1 + \frac{10^n}{9^n} \), shows the upward trend or rising nature of the sequence values as \( n \) increases. This growing pattern suggests a lack of convergence visually supporting the calculated results—demonstrating the sequence's divergence.
The value of graphical representation in calculus cannot be overstated. It provides immediate insight into numerical behaviors and trends that may not be immediately evident through calculations alone.