Problem 14 - 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 \(\lim _{n \rightarrow \infty} a_{n}\). $$ a_{n}=\left(\frac{1}{4}\right)^{n}+3^{n / 2} $$

Step-by-Step Solution

Answer

The sequence diverges and doesn't have a finite limit.

Step 1: Identify the Formula

The formula for the sequence is given as \(a_n = \left(\frac{1}{4}\right)^{n} + 3^{n/2}\). Our task is to find the first five terms of this sequence.

Step 2: Calculate the First Five Terms

Let's compute the first five terms one by one.- For \(n = 1\), \(a_1 = \left(\frac{1}{4}\right)^1 + 3^{1/2} = \frac{1}{4} + \sqrt{3}\).- For \(n = 2\), \(a_2 = \left(\frac{1}{4}\right)^2 + 3^{2/2} = \frac{1}{16} + 3\).- For \(n = 3\), \(a_3 = \left(\frac{1}{4}\right)^3 + 3^{3/2} = \frac{1}{64} + 3\sqrt{3}\).- For \(n = 4\), \(a_4 = \left(\frac{1}{4}\right)^4 + 3^{4/2} = \frac{1}{256} + 9\).- For \(n = 5\), \(a_5 = \left(\frac{1}{4}\right)^5 + 3^{5/2} = \frac{1}{1024} + 3\sqrt{3^2}\).

Step 3: Determine Convergence or Divergence

To find out if the sequence converges or diverges, inspect the two components: \(\left(\frac{1}{4}\right)^{n}\) and \(3^{n/2}\).- The term \(\left(\frac{1}{4}\right)^{n}\) approaches zero as \(n\) increases, since any fraction less than one raised to the power \(n\) tends to zero.- The term \(3^{n/2}\) grows exponentially larger since \(3\) is raised to an increasing exponent.Since \(3^{n/2}\) dominates and increases without bound as \(n\) grows large, the sequence \(a_n\) diverges.

Step 4: Conclusion on Limit

Based on our analysis in the previous step, \(a_n = \left(\frac{1}{4}\right)^n + 3^{n/2}\) diverges. Thus, \(\lim_{n \to \infty} a_n = \infty\), indicating the sequence does not converge to any finite limit.

Key Concepts

Explicit Formula

An explicit formula allows us to find any term in a sequence directly, without needing to know the term that comes before it. In the given exercise, the explicit formula is\[a_{n} = \left(\frac{1}{4}\right)^{n} + 3^{n/2},\]This formula is composed of two parts:

• The first part, \( \left(\frac{1}{4}\right)^{n} \), is a fractional term that decreases as \( n \) increases because the base is less than one.
• The second part, \( 3^{n/2} \), is a power term that becomes larger as \( n \) increases since its base is greater than one.

Together, these contribute to the overall behavior of the sequence by calculating individual sequence terms.

Sequence Terms

The sequence terms are individual values in the series derived using the explicit formula. Here, we calculate the first five sequence terms:

• For \( n = 1 \), the term is \( a_1 = \frac{1}{4} + \sqrt{3} \).
• For \( n = 2 \), the term is \( a_2 = \frac{1}{16} + 3 \).
• For \( n = 3 \), the term is \( a_3 = \frac{1}{64} + 3\sqrt{3} \).
• For \( n = 4 \), the term is \( a_4 = \frac{1}{256} + 9 \).
• For \( n = 5 \), the term is \( a_5 = \frac{1}{1024} + 3\cdot3 \).

As \( n \) increases, the first term, which is increasingly small, contributes less to the sequence term compared to the rapidly growing second term. This shows how the explicit formula translates into actual values.

Limit of a Sequence

The limit of a sequence refers to the value that the terms of the sequence approach as the sequence progresses to infinity. For any convergence analysis, it's crucial to determine if the sequence has a finite limit. In our case:

• As \( n \to \infty \), \( \left(\frac{1}{4}\right)^{n} \), a fractional term, approaches zero.
• Meanwhile, \( 3^{n/2} \), being an exponential term, increases indefinitely.

Since one term grows without limit, the sequence cannot settle on a finite value, leading us to conclude that there's no finite limit in this case.

Divergence Analysis

Divergence analysis evaluates whether a sequence tends toward infinity, remains bounded, or converges to a specific value. In this exercise, the analysis shows that:- The term \( \left(\frac{1}{4}\right)^{n} \) decreases and heads towards zero.- The term \( 3^{n/2} \) grows exponentially and dominates the behavior of the sequence.The exponential growth of \( 3^{n/2} \) implies that the sequence \( a_n \) diverges, without reaching any bound. As such, \[\lim_{n \to \infty} a_n = \infty,\]indicating an infinite and unbounded sequence. Understanding both terms’ behavior was crucial in this divergence analysis.