Problem 48 - Calculus: Early Transcendentals

Question

Determine whether the sequence converges or diverges. If it converges, find the limit. \( a_n = \sqrt[n]{n} \)

Step-by-Step Solution

Answer

The sequence converges to 1.

Step 1: Identify the Given Sequence

The sequence provided is \( a_n = \sqrt[n]{n} \). We need to determine if this sequence converges or diverges as \( n \) approaches infinity and find the limit if it converges.

Step 2: Rewrite the Sequence in Logarithmic Form

Express \( a_n = \sqrt[n]{n} = n^{1/n} \). Taking the natural logarithm, we have \( \ln(a_n) = \frac{1}{n}\ln(n) \). This simplifies the expression and helps to analyze the sequence.

Step 3: Evaluate the Limiting Behavior of the Logarithmic Sequence

Examine the limit \( \lim_{n \to \infty} \frac{\ln(n)}{n} \). Apply L'Hopital's rule to this \( \frac{0}{0} \) form: The derivative of \( \ln(n) \) is \( \frac{1}{n} \) and the derivative of \( n \) is 1. Thus, \( \lim_{n \to \infty} \frac{1/n}{1} = \lim_{n \to \infty} \frac{1}{n} = 0 \).

Step 4: Apply Exponential Function to Find Final Limit

Since \( \lim_{n \to \infty} \ln(a_n) = 0 \), by exponentiating both sides, we conclude \( \lim_{n \to \infty} a_n = e^0 = 1 \). Therefore, the sequence converges to 1.

Key Concepts

L'Hopital's Rule

L'Hopital's Rule is an essential tool in calculus for determining limits that arise in an indeterminate form such as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). Using this rule enables us to solve limits by differentiating the numerator and the denominator until a determinate form is reached. For example, consider the limit of the sequence \( \frac{\ln(n)}{n} \) as \( n \) approaches infinity. Initially, both the numerator and the denominator approach infinity, resulting in an indeterminate form. By applying L'Hopital's Rule:

• Differentiate the numerator: the derivative of \( \ln(n) \) is \( \frac{1}{n} \).
• Differentiate the denominator: the derivative of \( n \) is 1.

This transforms the original limit into \( \lim_{n \to \infty} \frac{1/n}{1} = \lim_{n \to \infty} \frac{1}{n} = 0 \). Once we achieve a determinate form, we can evaluate the limit more straightforwardly.
L'Hopital's Rule is especially useful in sequences and series analysis and is frequently applied to evaluate the limiting behavior of logarithms, exponential functions, and more.

Limit of a Sequence

The limit of a sequence is a fundamental concept in calculus, describing the value that a sequence approaches as the index goes to infinity.
In the case of scalars, if a sequence converges to a limit, it becomes arbitrarily close to this limit as the sequence progresses. Let's consider the sequence \( a_n = \sqrt[n]{n} = n^{1/n} \). The aim is to determine its convergence and limit:

• The limit process involves analyzing the expression \( \ln(n^{1/n}) = \frac{1}{n} \ln(n) \).
• The evaluation requires a method to confront infinity, hence utilizing L'Hopital's Rule for simplification is appropriate.

Ultimately, after simplification and evaluating the limiting behavior, we find that \( \lim_{n \to \infty} a_n = 1 \). This confirms that the sequence converges to the limit of 1.
Understanding how sequences behave as they progress is crucial in calculus, providing insights into inference patterns and continuity, and supporting broader mathematical reasoning.

Logarithmic Form

Converting a sequence into a logarithmic form is a powerful technique used to simplify complex functions for easier analysis, especially when dealing with power and exponential sequences.
In our example with the sequence \( a_n = n^{1/n} \), the first step is expressing it logarithmically: \( \ln(a_n) = \frac{1}{n} \ln(n) \). This transformation is essential because:

• Logarithmic operations often transform multiplication into addition or division into subtraction, simplifying analysis.
• It allows us to apply L'Hopital's Rule more effectively if indeterminate forms appear.

After simplifying \( \ln(a_n) \), and finding that \( \lim_{n \to \infty} \ln(a_n) = 0 \), we can conclude the behavior of the original sequence by using the exponential function: \( \lim_{n \to \infty} a_n = e^0 = 1 \).
Understanding and manipulating logarithmic forms requires practice but offers a clear path to evaluate limits and analyze convergence in a variety of sequences across calculus.