Problem 21 - 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 \(\lim _{n \rightarrow \infty} a_{n}\). $$ \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \ldots $$

Step-by-Step Solution

Answer

The sequence converges to 1 with an explicit formula \( a_n = \frac{n}{n+1} \).

Step 1: Identify the General Form

Observe the sequence: \( \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \ldots \). Notice that the numerator increases linearly by 1 each time, starting from 1, and the denominator increases similarly, starting from 2. So for the \(n\)-th term, the numerator is \(n\) and the denominator is \(n+1\).

Step 2: Write the General Formula

Using the observation from Step 1, we can express the \(n\)-th term of the sequence as:\[ a_n = \frac{n}{n+1} \]

Step 3: Determine Convergence or Divergence

To determine if this sequence converges or diverges, examine the behavior of \( a_n \) as \( n \) approaches infinity. We need to calculate the limit:\[ \lim_{n \to \infty} \frac{n}{n+1} \]

Step 4: Calculate the Limit

Simplify the fraction to find the limit:\[\lim_{n \to \infty} \frac{n}{n+1} = \lim_{n \to \infty} \left( 1 - \frac{1}{n+1} \right) = 1 - \lim_{n \to \infty} \frac{1}{n+1} = 1 - 0 = 1\]Hence, as \( n \to \infty \), the sequence term approaches 1.

Step 5: Conclude the Behavior

The sequence \( a_n = \frac{n}{n+1} \) converges, and the limit is \( \lim_{n \to \infty} a_n = 1 \).

Key Concepts

Sequences

A sequence is a list of numbers arranged in a special order. Each number in this list is called a term. In mathematics, sequences are usually defined by formulas that help describe how each term relates to its position in the sequence. For example, in our exercise, we have the sequence \( \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \ldots \). Here, each term is defined by its position in the list. The first term is \( \frac{1}{2} \), the second term is \( \frac{2}{3} \), and so on. It's often useful to use a letter, like 'n,' to stand for the position of a term. This makes it easy to describe sequences using formulas.

Explicit Formula

An explicit formula allows you to find any term in a sequence quickly, without first finding the previous terms. This is like having a shortcut. For our sequence \( \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \ldots \), we found the pattern for the \(n\)-th term: numerator is \(n\) and denominator is \(n+1\). With this pattern, we formulated the explicit formula: \[ a_n = \frac{n}{n+1} \]This formula tells us that for any position \(n\), we can find the corresponding term without counting one by one from the start. This makes your calculations faster and more efficient.

Convergence and Divergence

Sequences can behave differently as they grow larger. Some get closer and closer to a particular number, while others do not. This behavior is called convergence and divergence. - **Convergence:** The sequence approaches a single fixed value as it move towards infinity. - **Divergence:** The sequence does not approach any single value, but instead grows bigger, smaller, or oscillates.In the given problem, to check the nature of the sequence \( a_n = \frac{n}{n+1} \), we analyze its behavior as \(n\) becomes infinitely large. By calculating its limit, we can conclude if it converges or diverges.

Limits

The concept of limits helps us understand the behavior of sequences as they approach infinity. To find the limit of a sequence, we see what happens to the sequence's terms as they get closer to infinity. In our example, we calculated the limit for \( a_n = \frac{n}{n+1} \). By simplifying it: \[ \lim_{n \to \infty} \frac{n}{n+1} = \lim_{n \to \infty} \left( 1 - \frac{1}{n+1} \right) = 1 - \lim_{n \to \infty} \frac{1}{n+1} = 1 - 0 = 1 \]This means as \(n\) gets very large, the sequence terms get as close as we like to the number 1. Therefore, the sequence converges, and its limit is 1. Understanding limits is crucial to determining the eventual behavior of sequences.