Problem 30 - 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 \(\lim _{n \rightarrow \infty} a_{n}\). \(1-\frac{1}{2}, \frac{1}{2}-\frac{1}{3}, \frac{1}{3}-\frac{1}{4}, \frac{1}{4}-\frac{1}{5}, \ldots\)

Step-by-Step Solution

Answer

The sequence is \(a_n = \frac{1}{n} - \frac{1}{n+1}\), and it converges to 0.

Step 1: Analyze the sequence

The given sequence is \(1 - \frac{1}{2}, \frac{1}{2} - \frac{1}{3}, \frac{1}{3} - \frac{1}{4}, \frac{1}{4} - \frac{1}{5}, \ldots\). Notice the pattern of each term: \(a_n = \frac{1}{n} - \frac{1}{n+1}\).

Step 2: Find an explicit formula

We identified the pattern as \(a_n = \frac{1}{n} - \frac{1}{n+1}\). This is our explicit formula for the sequence.

Step 3: Determine convergence or divergence

To determine if the sequence converges, look at \(\lim _{n \rightarrow \infty} a_n\). Evaluate: \(\lim _{n \rightarrow \infty} \left(\frac{1}{n} - \frac{1}{n+1}\right)\).

Step 4: Calculate the limit at infinity

As \(n\) approaches infinity, both \(\frac{1}{n}\) and \(\frac{1}{n+1}\) approach zero. Thus, \(\lim _{n \rightarrow \infty} \left( \frac{1}{n} - \frac{1}{n+1} \right) = 0\).

Step 5: Conclude convergence

Since \(\lim _{n \rightarrow \infty} a_n = 0\), the sequence converges to 0.

Key Concepts

Explicit Formula for Sequences

Sequences are ordered lists of numbers, and for many sequences, determining an explicit formula can help identify the general term. In this exercise, the sequence presented is: \[1 - \frac{1}{2}, \frac{1}{2} - \frac{1}{3}, \frac{1}{3} - \frac{1}{4}, \frac{1}{4} - \frac{1}{5}, \ldots \] To find the formula, notice the pattern of each term. For the given sequence, the expression resembles a telescoping series. Each term can be defined by the formula:

• \(a_n = \frac{1}{n} - \frac{1}{n+1}\)

"\(a_n\)" is the general term, which allows you to generate any term in the sequence by substituting "n" with the term number. This formula simplified the relationship between consecutive terms, making the sequence easier to work with. Once we have this explicit formula, applying it can assist in analyzing deeper aspects like convergence.

Sequence Convergence

Convergence is an important property in sequences, indicating whether the sequence approaches a specific value as it unfolds through its terms. To determine if a sequence like \(a_n = \frac{1}{n} - \frac{1}{n+1}\) converges, it's necessary to evaluate the behavior as \(n\) increases indefinitely. Here's how to check for convergence:

• Examine \(\lim_{n \to \infty} a_n\): what happens to the terms as \(n\) gets very large?

For our sequence, we specifically look at:

• \(\lim_{n \to \infty} \left(\frac{1}{n} - \frac{1}{n+1}\right)\)

Both \(\frac{1}{n}\) and \(\frac{1}{n+1}\) approach zero as \(n\) increases. Because the two components vanish at the rate, their difference also vanishes, meaning the sequence converges. This confirms that the sequence settles towards a specific point.

Limit of a Sequence

The limit of a sequence is the value that the terms of the sequence approach as the term number becomes very large. For the sequence \(a_n = \frac{1}{n} - \frac{1}{n+1}\), the calculation of its limit at infinity provides insight into the sequence's ultimate behavior. To compute the limit:

• Evaluate: \(\lim_{n \to \infty} \left(\frac{1}{n} - \frac{1}{n+1}\right)\).
• Both terms \(\frac{1}{n}\) and \(\frac{1}{n+1}\) shrink to zero as \(n\) becomes very large.

The principle of limits tells us that when each part of an expression goes to zero, their difference also zeros out. Thus, for this sequence:

• \(\lim_{n \to \infty} a_n = 0\)

As such, the sequence converges, and its limit is 0. Understanding the limit helps predict the long-term behavior and stability of the sequence, which is especially crucial in both pure and applied mathematics.