Problem 21 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 21-30, find an explicit formula \(a_{n}=\) for each sequence, determine whether the sequence converges or diverges, and, if it converges, find \(\lim _{n \rightarrow \infty} a_{n}\). \(\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \ldots\)

Step-by-Step Solution

Answer

The sequence has an explicit formula \(a_n = \frac{n}{n+1}\), converges, with a limit of 1.

Step 1: Identify the Pattern

Observe the sequence: \(\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \ldots\). Note that each term is consecutive integers increasing in both numerator and denominator. Specifically, the numerator is \(n\) while the denominator is \(n+1\).

Step 2: Write the Explicit Formula

Since we identified from the pattern that each numerator is \(n\) and the denominator is \(n+1\), the explicit formula for the sequence is \(a_n = \frac{n}{n+1}\).

Step 3: Analyze Convergence

To determine if the sequence converges, find the limit as \(n\) approaches infinity: \(\lim_{n \to \infty} \frac{n}{n+1}\). Simplifying, it becomes \(\lim_{n \to \infty} \frac{1}{1+\frac{1}{n}}\). As \(n\) approaches infinity, \(\frac{1}{n}\) approaches 0, thus the expression approaches \(\frac{1}{1+0} = 1\).

Step 4: Conclusion

The sequence converges to 1 as \(n\) approaches infinity. Therefore, we conclude that \(\lim_{n \to \infty} a_n = 1\).

Key Concepts

Explicit Formula

In mathematics, an explicit formula is a direct expression of the terms of a sequence using a simple formula. It allows us to compute any term without needing the previous term's value. Let's explore this with the sequence given: \( \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \ldots \).To find the explicit formula for this sequence, we need to identify a pattern. Here, we see that the numerator of each term increases by one, starting at 1, then 2, 3, and so on. The denominator follows a similar pattern, but each value is one more than the numerator. So, if the numerator is \( n \), the denominator is \( n+1 \).Thus, the explicit formula for the \( n \)-th term in the sequence becomes \( a_n = \frac{n}{n+1} \). This formula allows us to find any term in the sequence by simply plugging in the value of \( n \), making it highly efficient for computations. This concept is foundational in sequences, offering a simple way to explore their behavior.

Limit of a Sequence

The limit of a sequence is a crucial concept in understanding the behavior of sequences as they progress towards infinity. It asks what value the terms settle upon or approach as the index \( n \) becomes very large.For the sequence with the explicit formula \( a_n = \frac{n}{n+1} \), we need to determine the limit as \( n \to \infty \). To do this, examine the formula:\[ \lim_{n \to \infty} \frac{n}{n+1} = \lim_{n \to \infty} \frac{1}{1+\frac{1}{n}} \]As \( n \) grows larger, the term \( \frac{1}{n} \) becomes very small, nearing zero. Hence, the expression simplifies to \( \frac{1}{1+0} = 1 \). This tells us that as \( n \) increases, the sequence \( a_n \) approaches 1, indicating convergence.Understanding limits is vital as it shows whether a sequence stabilizes at a certain point or continues to wander off indefinitely. It tells us about the long-term behavior of sequences, providing insights into stability and convergence.

Infinite Series

While this exercise focuses on the convergence of a sequence, it indirectly connects us to the concept of infinite series. An infinite series is the sum of the terms of an infinite sequence.For example, if we were to sum the terms of a sequence like \( \frac{1}{2} + \frac{2}{3} + \frac{3}{4} + \ldots \), it forms an infinite series. To determine whether this infinite sum converges or diverges, we would explore methods such as the comparison test or ratio test to analyze the series behavior.Even though the sequence \( a_n = \frac{n}{n+1} \) itself converges to 1, summing its terms forms a different type of analysis. Not every sequence that converges will lead to a convergent series once summed, and this nuanced distinction highlights the rich complexity within infinite sequences and series.By understanding sequences and their limits, one builds a foundation that's essential for tackling more advanced topics like series. Series unlocks explorations of areas, volumes, and other infinite summations, powering many aspects of calculus and mathematical analysis.