Problem 57 - Thomas Calculus

Question

Which of the sequences \(\left\\{a_{n}\right\\}\) in Exercises \(27-90\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\left(\frac{3}{n}\right)^{1 / n} $$

Step-by-Step Solution

Answer

The sequence \(a_{n} = \left(\frac{3}{n}\right)^{1/n}\) converges with a limit of 1.

Step 1: Determine the Form of the Sequence

The given sequence is \(a_{n} = \left( \frac{3}{n} \right)^{1/n}\). We need to determine whether this sequence converges or diverges as \(n \to \infty\). Our goal is to find the limit \(\lim_{{n \to \infty}} a_{n}\).

Step 2: Recognize Known Limits and Properties

Recall the property that \(\lim_{{n \to \infty}} n^{1/n} = 1\). This property applies since \((\frac{3}{n})\) can be rewritten as a product of a constant \(3^{1/n}\) and \(n^{1/n}\). This way, \(a_n\) can be expanded as \((\frac{3^{1/n}}{n^{1/n}})\).

Step 3: Simplify Using Exponentiation

Simplify the expression using the property \(n^{1/n} = e^{\ln(n)/n}\). Then the sequence \(a_n = \left(\frac{3}{n}\right)^{1/n} = \frac{3^{1/n}}{n^{1/n}}\). Analyze: \(3^{1/n} \to 1\) and \(n^{1/n} \to 1\) as \(n \to \infty\).

Step 4: Calculate the Limit

Now compute the limit: \(\lim_{{n \to \infty}} a_{n} = \lim_{{n \to \infty}} \frac{3^{1/n}}{n^{1/n}} = \frac{\lim_{{n \to \infty}} 3^{1/n}}{\lim_{{n \to \infty}} n^{1/n}} = \frac{1}{1} = 1\).

Step 5: Conclude Convergence

Because the limit of the sequence \(a_n\) as \(n\) approaches infinity is \(1\), the sequence converges.

Key Concepts

Convergent Sequences

A sequence is said to be convergent if its terms approach a specific finite value as the index goes to infinity. In simpler terms, as you move further along in the sequence, the numbers get closer and closer to a particular number. Understanding convergent sequences is essential in calculus as they help in understanding functions and sums.

• Each term in the sequence becomes arbitrarily close to a fixed number, known as the limit.
• Convergent sequences play a critical role in defining integrals and derivatives in calculus.

For example, consider the sequence provided in the exercise, \(a_n = \left( \frac{3}{n} \right)^{1/n}\). As \(n\) increases, this sequence approaches the limit \(1\). This behavior indicates that our sequence is convergent.

Limits of Sequences

The limit of a sequence is the value the terms of the sequence approach as the index becomes very large. If a sequence converges, it means that this limit exists. In calculus, finding the limit of sequences is crucial for analyzing their behavior.

• The expression \(\lim_{{n \to \infty}} a_n\) denotes the limit of the sequence.
• For a limit to exist, the sequence must steadily get closer to a particular value when the index \(n\) is very large.

Using the exercise as an example, for the sequence \(a_n = \left( \frac{3}{n} \right)^{1/n}\), we simplified and identified its limit as \(1\). By investigating the expression as \(n\) approaches infinity, we were able to determine this behavior.

Properties of Sequences

Understanding the fundamental properties of sequences helps in determining the behavior of the sequence and predicting its convergence or divergence. Several key properties can be identified in sequences:

• Monotonicity: A sequence is monotonic if it is either entirely non-increasing or non-decreasing.
• Boundedness: A sequence is bounded if there is a finite number that the sequence's terms do not exceed.
• Behavior at Infinity: As shown in this example, simple transformations, like taking roots or exponentiating, help to dictate how sequences behave as \(n\rightarrow\infty\).

In the case of the given sequence, \(a_n = \left( \frac{3}{n} \right)^{1/n}\), the sequence exhibits behavior that allows its limit to be evaluated as \(n\) becomes large using principles such as the limit of \(n^{1/n}\) approaching \(1\). Knowing these properties helps solve similar problems with ease.