Problem 64 - Essential Calculus

Question

Determine whether the sequence with the given \(n\) th term is monotonic. Discuss the boundedness of the sequence. Use a graphing utility to confirm your results. \(a_{n}=\frac{3 n}{n+2}\)

Step-by-Step Solution

Answer

The sequence \(a_{n}=\frac{3 n}{n+2}\) is an increasing (thus monotonic) and bounded sequence. The lower bound of the sequence is 0 and the upper bound is 3.

Step 1: Determine the Monotonicity

For a sequence to be monotonic, it must be entirely non-increasing or non-decreasing. Their difference \(a_{n+1} - a_n\) can indicate whether the sequence is increasing or decreasing. If \(a_{n+1} - a_n > 0\) then the sequence is increasing. If \(a_{n+1} - a_n < 0\), then the sequence is decreasing.

Step 2: Computed the Difference \(a_{n+1} - a_{n}\)

Substitute \(n+1\) for \(n\) in the given sequence \(a_{n}\) to get \(a_{n+1}=\frac{3(n+1)}{(n+1)+2}=\frac{3n+3}{n+3}\), then find the difference. \(a_{n+1} - a_{n} =\frac{3n+3}{n+3} - \frac{3n}{n+2}\). Solve this difference, the result is: \(a_{n+1} - a_{n} = \frac{6}{(n+2)(n+3)}\). Since \(n >= 1\), the difference is always positive, hence the sequence is increasing.

Step 3: Determine the Boundedness of the Sequence

The sequence is bounded if there are upper or lower limits. It is clear that the sequence is positive for all values of \(n >= 1\), hence it is lower bounded by 0. Next, if you take the limit of the sequence as \(n\) approaches infinity, i.e. \(\lim_{n \to \infty} \frac{3n}{n+2} =3\), it is clear that the sequence is also upper bounded by 3.

Step 4: Confirm the Results with a Graphing Utility

Plotting the sequence with a graphing utility will help confirm the results. The x-axis can be considered as 'n' (the term number) and the y-axis represents the sequence \(a_{n}\). The plot should indicate the sequence as increasing and bounded by the lines \(y=0\) (lower bound) and \(y=3\) (upper bound).

Key Concepts

Sequence Boundedness

In mathematics, a sequence is said to be bounded if it has both a lower and an upper bound. For the sequence given by the terms \( a_n = \frac{3n}{n+2} \), we can determine its boundedness by assessing its limits.
First, consider the lower bound. Since both the numerator and denominator of the fraction \( \frac{3n}{n+2} \) are positive for \( n \geq 1 \), the value of each term \( a_n \) is also positive. Thus, the sequence is at least lower bounded by 0.
Next, evaluate the upper bound by taking the limit of the sequence as \( n \) approaches infinity. The expression \( \frac{3n}{n+2} \) simplifies to \( 3 \) when evaluated in the limit, making 3 the upper bound of the sequence.
Therefore, our sequence is bounded with an interval from 0 to 3. This concept of boundedness is important because it helps us predict and understand the behavior of sequences over an infinite number of terms.

Calculus Sequences

Calculus sequences involve the application of limits and calculus concepts to analyze sequences. In this case, we are examining the sequence defined by \( a_n = \frac{3n}{n+2} \).
Sequences in calculus are often evaluated for convergence and monotonicity. Convergence pertains to whether a sequence approaches a particular limit as \( n \) goes to infinity. In this sequence, as \( n \) increases, \( \frac{3n}{n+2} \) tends towards 3. Thus, we say this sequence converges to the value of 3.
To check monotonicity, we look at the difference between consecutive terms, \( a_{n+1} - a_n \). If all the differences are positive for a given range of \( n \), the sequence is increasing, which was confirmed in our calculations: the difference \( \frac{6}{(n+2)(n+3)} \) is positive for \( n \geq 1 \). Hence, the sequence is strictly increasing. Understanding these properties allows us to analyze the nature of the sequence comprehensively.

Graphing in Calculus

Graphically representing sequences is an effective way to visually confirm their properties such as boundedness and monotonicity. For the sequence \( a_n = \frac{3n}{n+2} \), plotting the sequence helps us see its behavior across various values of \( n \).
When you plot \( a_n \) on a graph using a graphing utility, the x-axis represents \( n \), the term number, and the y-axis represents the value of each term.
As the sequence is graphed, you would notice the curve steadily increasing and gradually leveling off towards the upper bound of 3 as \( n \) increases. Additionally, the sequence never falls below 0, reiterating its lower bound.
Graphing not only supports the analytical results from calculus but also provides a clear and intuitive understanding of how the sequence behaves, making it a valuable tool for students learning calculus.