Problem 1 - Calculus Early Transcendentals: Pearson New International Edition

Question

An explicit formula for \(a_{n}\) is given. Write the first five terms of \(\left\\{a_{n}\right\\}\), determine whether the sequence converges or diverges, and, if it converges, find \(\lim _{n \rightarrow \infty} a_{n}\). $$ a_{n}=\frac{n}{3 n-1} $$

Step-by-Step Solution

Answer

The sequence converges, and the limit is \( \frac{1}{3} \).

Step 1: Calculate the first term

Use the explicit formula \( a_n = \frac{n}{3n-1} \) to find \( a_1 \). Substitute \( n=1 \) into the formula: \[ a_1 = \frac{1}{3(1) - 1} = \frac{1}{2} \]

Step 2: Calculate the second term

Find \( a_2 \) by substituting \( n=2 \) into the formula:\[ a_2 = \frac{2}{3(2) - 1} = \frac{2}{5} \]

Step 3: Calculate the third term

Find \( a_3 \) by substituting \( n=3 \) into the formula:\[ a_3 = \frac{3}{3(3) - 1} = \frac{3}{8} \]

Step 4: Calculate the fourth term

Find \( a_4 \) by substituting \( n=4 \) into the formula:\[ a_4 = \frac{4}{3(4) - 1} = \frac{4}{11} \]

Step 5: Calculate the fifth term

Find \( a_5 \) by substituting \( n=5 \) into the formula:\[ a_5 = \frac{5}{3(5) - 1} = \frac{5}{14} \]

Step 6: Write the first five terms

The first five terms of the sequence are: \( \frac{1}{2}, \frac{2}{5}, \frac{3}{8}, \frac{4}{11}, \frac{5}{14} \).

Step 7: Determine convergence or divergence

To determine if the sequence converges, find the limit as \( n \to \infty \). Calculate: \[ \lim_{n \to \infty} \frac{n}{3n-1} = \lim_{n \to \infty} \frac{1}{3 - \frac{1}{n}}. \]Since \( \frac{1}{n} \to 0 \) as \( n \to \infty \), the expression simplifies to \( \frac{1}{3} \). Thus, the sequence converges.

Step 8: Find the limit

The limit of the sequence as \( n \to \infty \) is \( \frac{1}{3} \).

Key Concepts

Convergence and Divergence

Understanding whether a sequence converges or diverges is crucial in calculus, as it tells us the behavior as we move towards infinity. In simple terms:

• A convergent sequence approaches a specific value as the term number increases indefinitely.
• If a sequence doesn't settle toward any particular number, it is considered divergent.

For the sequence given by the formula \( a_n = \frac{n}{3n-1} \), we check if this sequence converges or diverges by finding the limit as \( n \to \infty \). If the limit results in a finite value, the sequence converges; otherwise, it diverges. In the steps provided, we calculated that this sequence indeed converges because the limit approaches \( \frac{1}{3} \). This tells us that as \( n \) becomes extremely large, the terms in the sequence get very close to \( \frac{1}{3} \), confirming convergence.

Sequence Limit

The limit of a sequence gives us the end behavior of the sequence's terms as we move towards infinity. When talking about limits, especially in sequences, we're interested in what happens when \( n \) is very large.
To find the limit of the sequence \( a_n = \frac{n}{3n-1} \), we apply our knowledge of limits.

• We treated large \( n \) by simplifying the expression: \( \lim_{n \to \infty} \frac{n}{3n-1} = \lim_{n \to \infty} \frac{1}{3 - \frac{1}{n}} \).
• Recognizing that \( \frac{1}{n} \) approaches 0 as \( n \) increases infinitely, the expression simplifies to \( \frac{1}{3} \).

So, the limit is \( \frac{1}{3} \), indicating that the farther we go, the sequence behaves increasingly like this value. Understanding sequence limits helps to determine convergence and the precise value that a converging sequence approaches.

Explicit Formula for Sequences

An explicit formula in sequences is a way of directly expressing the \( n^{th} \) term, \( a_n \), as a function of \( n \). This allows us to calculate any term without knowing the previous term.
For our sequence, the explicit formula given is \( a_n = \frac{n}{3n-1} \). This formula is significant because:

• It enables quick computation of any term in the sequence. For example, using it, we easily calculated the first five terms as \( \frac{1}{2}, \frac{2}{5}, \frac{3}{8}, \frac{4}{11}, \frac{5}{14} \).
• It aids in analyzing the general behavior of the sequence as \( n \) increases, which was critical in determining the convergence and ultimate limit of \( \frac{1}{3} \).

Thus, having an explicit formula is exceptionally helpful in both practical calculations and theoretical analysis in sequences.