Problem 42 - Essential Calculus

Question

Determine the convergence or divergence of the sequence with the given \(n\) th term. If the sequence converges, find its limit. \(a_{n}=\frac{\ln \sqrt{n}}{n}\)

Step-by-Step Solution

Answer

The given sequence \(a_{n}=\frac{\ln{\sqrt{n}}}{n}\) converges and its limit is \(0\).

Step 1: Preliminary Mathematics

To check the convergence of \(a_{n}\), we need to find \(\lim_{{n \to \infty}}\frac{\ln \sqrt{n}}{n}\). Before we do this, a property of logarithms can be used that states \(\ln a^k = k \ln a\). Therefore, we can write the sequence as \(a_{n}=2\cdot\frac{\ln{n}}{n}\).

Step 2: L'Hopital’s Rule

Next, it's important to note that this form is similar to \(\lim_{{x \to a}}\frac{f(x)}{g(x)}\), which is an indeterminate form of type 0/0 or ∞/∞. We can make use of L'Hopital’s rule here, which states that this limit is the same as the limit of the ratios of their derivatives. We differentiate the numerator and denominator separately, which gives us \(\frac{d}{d n}(\ln n) = 1/n\), \(\frac{d}{d n}n = 1\). Applying L'Hopital’s rule, we get \(\lim_{{n \to \infty}}\frac{\ln{n}}{n} = \lim_{{n \to \infty}}\frac{1/n}{1}\), which simplifies to \(\lim_{{n \to \infty}}(\frac{1}{n})\).

Step 3: Finding the limit

Find the limit of \(\frac{1}{n}\) as \(n\) approaches infinity. The result is \(0\). Hence, the sequence \(a_{n}=\frac{\ln{\sqrt{n}}}{n}\) converges and its limit is \(0\).

Key Concepts

L'Hopital’s Rule

L'Hopital’s Rule provides a remarkable method for calculating limits, especially when we encounter an indeterminate form like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). When a limit fits these forms, you can apply L'Hopital’s Rule to evaluate it more easily.
To use L'Hopital’s Rule, you need to:

• Identify that the limit is an indeterminate form.
• Differentiate the numerator and denominator separately.
• Re-evaluate the limit using these derivatives.

In our exercise, we started with the limit of \( \lim_{{n \to \infty}} \frac{\ln \sqrt{n}}{n} \). By simplifying with a logarithmic property, it turned into \( \lim_{{n \to \infty}} 2 \cdot \frac{\ln n}{n} \).
We noticed that this is of form \( \frac{\infty}{\infty} \), so applying L'Hopital’s Rule involved differentiating the numerator (\( \ln n \) becomes \( \frac{1}{n} \)) and the denominator (\( n \) becomes \( 1 \)).
This gave us a new limit to solve: \( \lim_{{n \to \infty}} \frac{1/n}{1} \), which simplifies to \( \lim_{{n \to \infty}} \frac{1}{n} \), leading to our solution.

Limit of a Sequence

The limit of a sequence is foundational in determining whether a sequence converges or diverges. A sequence converges if it approaches a particular value as the index \( n \) approaches infinity. If no such value exists, the sequence diverges.
In simpler terms, as you keep adding more and more terms of the sequence, if they keep getting closer to a specific number, you have convergence. If not, you're dealing with divergence.
In the given exercise, we found the limit \( \lim_{{n \to \infty}} \frac{1}{n} \). This expression goes to \( 0 \) as \( n \) becomes very large because \( \frac{1}{n} \) becomes smaller and smaller, essentially getting closer to zero.
Thus, our sequence \( a_{n}=\frac{\ln{\sqrt{n}}}{n} \) converges, and its limit is \( 0 \). This understanding allows us to determine behavior at infinity, a core aspect of analysis.

Logarithmic Functions

Logarithmic functions, represented by \( \ln(x) \), have unique properties that can simplify complex expressions. One key property is \( \ln(a^k) = k\ln(a) \). This allows us to simplify terms inside the logarithm.
In our exercise, the sequence was \( a_{n}=\frac{\ln \sqrt{n}}{n} \). Applying the property \( \ln (\sqrt{n}) = \frac{1}{2}\ln(n) \) helped rewrite the sequence in a simpler form, \( a_{n}=\frac{\ln n}{2n} \).
These properties:

• Allow transformation and simplification of logarithmic expressions.
• Make it easier to apply other mathematical tools, like L'Hopital’s Rule.

Understanding logarithmic functions can thus make complex calculations more approachable, as seen by simplifying before applying techniques like L'Hopital’s Rule. These transformations are pivotal in evaluating limits effectively.