Problem 26 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

The explicit formula is \( a_n = \frac{1 + \frac{1}{n}}{1 + \frac{2}{n}} \), and the sequence converges to 1.

Step 1: Identifying the General Term (Denominator)

The given sequence has terms in the form \( \frac{n}{n+1 - \frac{1}{n+1}} \). We'll focus on simplifying the denominator. The denominator for the \( n^{th} \) term is \( n+1 - \frac{1}{n+1} \).

Step 2: Simplifying the Denominator

The expression \( n+1 - \frac{1}{n+1} \) can be rewritten by getting a single fraction: \( \frac{(n+1)^2 - 1}{n+1} \). This simplifies to \( \frac{n^2 + 2n}{n+1} \).

Step 3: Formulating the Explicit Formula

Now, substitute the simplified denominator back into the sequence's term: \( a_n = \frac{n}{\frac{n^2 + 2n}{n+1}} \). Simplify this as \( a_n = \frac{n(n+1)}{n^2 + 2n} = \frac{n^2 + n}{n^2 + 2n} \).

Step 4: Further Simplifying the Explicit Formula

The expression \( \frac{n^2 + n}{n^2 + 2n} \) can be simplified by dividing every term by \( n^2 \): \( \frac{1 + \frac{1}{n}}{1 + \frac{2}{n}} \).

Step 5: Finding the Limit as n Approaches Infinity

As \( n \to \infty \), the terms \( \frac{1}{n} \) and \( \frac{2}{n} \) approach 0. Thus, \( \lim_{n \to \infty} a_n = \frac{1+0}{1+0} = 1 \).

Step 6: Convergence or Divergence

Since \( \lim_{n \to \infty} a_n = 1 \), the sequence converges to 1.

Key Concepts

Explicit Formula for a Sequence

An explicit formula for a sequence is an equation that allows you to calculate any term in the sequence just by plugging in the position, or the index, of the term. This position is often denoted by \( n \). For the given sequence, we want to find an explicit equation for the terms: \( a_n = \frac{n}{n+1 - \frac{1}{n+1}} \). Finding this equation involves simplifying the expression for the denominator.
First, observe that each fraction in the sequence comprises a numerator \( n \) and a denominator \( n+1 - \frac{1}{n+1} \). Expanding the denominator by combining into a single fraction gives us the generalized form:

• Denominator: \( \frac{(n+1)^2 - 1}{n+1} = \frac{n^2 + 2n}{n+1} \)

Hence, substituting back into the larger fraction results in:
\( a_n = \frac{n(n+1)}{n^2 + 2n} \), which simplifies further to \( \frac{n^2 + n}{n^2 + 2n} \). Having a clear explicit formula enables you to compute any desired term directly.

Convergence and Divergence

To determine if a sequence converges or diverges, we examine if the sequence approaches a specific number as \( n \) becomes extremely large. Convergence means the sequence gets closer to a fixed number, known as the limit. Divergence indicates the sequence does not approach any specific limit and might increase indefinitely or oscillate.
For our sequence \( a_n = \frac{n^2 + n}{n^2 + 2n} \), we suspect a convergence because of the stable structure of its fractions when simplified for large \( n \). If a sequence converges, finding the limit requires observing the behavior of its terms as \( n \) tends toward infinity.
In practice, if simplifying leads to a form where variables diminish (approach zero), like \( \frac{1}{n} \), the sequence likely converges. Here, doing so, as shown in the following sections, verifies convergence to a specific number.

Sequence Simplification

Simplification is important for understanding the general behavior of sequences, especially as they grow larger. Reducing each term to its simplest form can make it easier to see if and where convergence or divergence occurs.
Let’s break down the given sequence: \( a_n = \frac{n^2 + n}{n^2 + 2n} \). Our goal is to simplify this in preparation for limit analysis. Begin by simplifying the terms by dividing each by \( n^2 \). You'll get:

• Simplified form: \( \frac{1 + \frac{1}{n}}{1 + \frac{2}{n}} \)

Notice the terms \( \frac{1}{n} \) and \( \frac{2}{n} \) vanish as \( n \) approaches infinity. This simplification helps to see the convergence potential and serves as a precursor to calculating the sequence’s limit.

Limit Calculation

Calculating the limit of a sequence is a key step in determining its behavior as \( n \) becomes very large. This tells you where the sequence eventually "settles," if it does at all.
For the sequence described by \( a_n = \frac{n^2 + n}{n^2 + 2n} \), now in its simplified form \( \frac{1 + \frac{1}{n}}{1 + \frac{2}{n}} \), you evaluate the limit as \( n \to \infty \):

• The term \( \frac{1}{n} \rightarrow 0 \)
• Similarly, \( \frac{2}{n} \rightarrow 0 \)
• Thus, \( \lim_{n \to \infty} a_n = \frac{1+0}{1+0} = 1 \)

This result indicates the sequence converges to 1. Understanding limit calculation helps grasp the end behavior of sequences, ensuring a thorough grasp of whether they converge or diverge.