Problem 4 - 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 \(\lim _{n \rightarrow \infty} a_{n}\). $$ a_{n}=\frac{3 n^{2}+2}{2 n-1} $$

Step-by-Step Solution

Answer

The sequence diverges.

Step 1: Calculate the First Term

To find the first term of the sequence, substitute \(n=1\) into the explicit formula: \(a_1= \frac{3(1)^2 + 2}{2(1) - 1} = \frac{3\cdot1 + 2}{2 - 1} = \frac{5}{1} = 5\). So, the first term is 5.

Step 2: Calculate the Second Term

For the second term, substitute \(n=2\) into the formula: \(a_2= \frac{3(2)^2 + 2}{2(2) - 1} = \frac{3\cdot4 + 2}{4 - 1} = \frac{14}{3}\). The second term is \(\frac{14}{3}\).

Step 3: Calculate the Third Term

For the third term, substitute \(n=3\): \(a_3= \frac{3(3)^2 + 2}{2(3) - 1} = \frac{3\cdot9 + 2}{6 - 1} = \frac{29}{5}\). The third term is \(\frac{29}{5}\).

Step 4: Calculate the Fourth Term

Substitute \(n=4\): \(a_4 = \frac{3(4)^2 + 2}{2(4) - 1} = \frac{3\cdot16 + 2}{8 - 1} = \frac{50}{7}\). The fourth term is \(\frac{50}{7}\).

Step 5: Calculate the Fifth Term

Substitute \(n=5\): \(a_5 = \frac{3(5)^2 + 2}{2(5) - 1} = \frac{3\cdot25 + 2}{10 - 1} = \frac{77}{9}\). The fifth term is \(\frac{77}{9}\).

Step 6: Determine Convergence or Divergence

To determine convergence, evaluate \(\lim_{n \to \infty} a_n = \lim_{n \to \infty} \frac{3n^2 + 2}{2n - 1}\). Considering the highest powers of \(n\) in the numerator and denominator, the expression simplifies to \(\frac{3n^2}{2n} = \frac{3n}{2}\). Thus, as \(n\) approaches infinity, \(a_n\) approaches infinity. Therefore, the sequence diverges.

Key Concepts

Explicit Formula

An explicit formula is a mathematical expression that allows the direct computation of any term in a sequence. For our given sequence \(a_n = \frac{3n^2 + 2}{2n-1}\), the explicit formula makes it easy to calculate the value of any term \(a_n\) by simply substituting the desired term number \(n\) into the equation.

• To compute the first term \(a_1\) of this sequence, we substitute \(n = 1\) into the formula, which yields \(a_1 = 5\).
• For \(a_2\), substitute \(n = 2\) to get \(a_2 = \frac{14}{3}\).
• Continue by substituting now \(n = 3, 4,\) and \(5\) for subsequent terms.
• This method provides a straightforward approach to compute each of the first five terms specifically: 5, \(\frac{14}{3}\), \(\frac{29}{5}\), \(\frac{50}{7}\), and \(\frac{77}{9}\).

By using the explicit formula, we can easily identify patterns or behaviors exhibited by the sequence.

Limit of a Sequence

A fundamental aspect of sequences in mathematics is understanding their behavior as the term number \(n\) grows indefinitely. The limit of a sequence \(a_n\) is denoted as \(\lim_{n \to \infty} a_n\) and describes the value that the sequence approaches as \(n\) becomes infinitely large.
For the given sequence, we calculated:

• \(\lim_{n \to \infty} \frac{3n^2 + 2}{2n - 1}\).
• To find this limit, look at the term with the highest degree of \(n\) in both the numerator and the denominator, which gives us \(\frac{3n^2}{2n}\).
• This simplifies to \(\frac{3n}{2}\).
• As \(n\) approaches infinity, \(\frac{3n}{2}\) also approaches infinity.

Thus, the sequence doesn't settle into a pattern and instead continues to grow larger indefinitely.

Sequence Convergence

In mathematical terms, a sequence is said to converge when its terms approach a specific value as \(n\) goes to infinity. If there's no such fixed value and the sequence keeps growing without bound, it diverges. To determine if a sequence converges, we primarily investigate its limit.
Consider the sequence \(a_n = \frac{3n^2 + 2}{2n - 1}\):

• The limit was found to be infinity, i.e., \(\lim_{n \to \infty} a_n = \infty\).
• Since the sequence continually increases without approaching any single finite value, this signifies divergence.
• A convergent sequence would have \(\lim_{n \to \infty} a_n = L\), where \(L\) would be a specific finite number.

Understanding the convergence characteristics helps in predicting the behavior of a sequence over time.

Infinite Limit

An infinite limit occurs when the terms of a sequence grow beyond all bounds as \(n\) increases indefinitely. This means the sequence doesn't approach a finite value, leading to divergence instead of convergence.
For the sequence \(a_n = \frac{3n^2 + 2}{2n - 1}\), let's examine:

• The calculation showed \(\lim_{n \to \infty} a_n\) is infinity.
• This result indicates that as \(n\) grows, \(a_n\) does not settle but keeps increasing.
• This type of sequence behavior is typical when the growth rate of the numerator is greater than that of the denominator in rational expressions.

Being able to recognize when a sequence has an infinite limit helps differentiate between diverging and converging behaviors in sequences.