Question

Determine the limit of the sequence or show that the sequence diverges. If it converges, find its limit. $$ a_{n}=\frac{\ln \left(n^{2}\right)}{\ln (2 n)} $$

Short answer

The sequence converges with a limit of 2.

Step-by-step solution

Simplify the Sequence Formula

Begin by simplifying the given sequence \(a_n = \frac{\ln(n^2)}{\ln(2n)}\). Use the property of logarithms that \(\ln(n^2) = 2\ln(n)\). The sequence becomes \(a_n = \frac{2\ln(n)}{\ln(2n)}\).

Apply the Logarithm Property

Apply the property of logarithms to the denominator: \(\ln(2n) = \ln(2) + \ln(n)\). Substitute this into the sequence to get \(a_n = \frac{2\ln(n)}{\ln(2) + \ln(n)}\).

Analyze the Limit as \(n\) Approaches Infinity

As \(n\) increases towards infinity, the term \(\ln(n)\) dominates over the constant \(\ln(2)\). Thus, the sequence \(a_n\) simplifies to \(a_n \approx \frac{2\ln(n)}{\ln(n)} = 2\), as the \(\ln(2)\) becomes negligible.

Determine the Limit of the Sequence

Since the sequence simplifies to approximately \(2\) for large \(n\), we conclude that the sequence converges and its limit is \(2\). Therefore, the limit \(\lim_{{n \to \infty}} a_n = 2\).

Key concepts

Logarithmic Properties

Logarithmic properties are essential tools in simplifying and manipulating expressions involving logarithms. They help us rewrite complex logarithmic expressions into simpler or more manageable forms. Here are several key properties of logarithms that are often useful:

• Power Rule: For any positive number \(b\) and any real number \(x\), \( \ln(b^x) = x \cdot \ln(b) \). This property helps us simplify expressions where the logarithm has a base with an exponent, as in the given exercise where \( \ln(n^2) = 2\ln(n) \).
• Product Rule: The logarithm of a product is the sum of the logarithms: \( \ln(ab) = \ln(a) + \ln(b) \). In our exercise, this rule is used to simplify the denominator \( \ln(2n) = \ln(2) + \ln(n) \).
• Quotient Rule: The logarithm of a quotient is the difference of the logarithms: \( \ln(a/b) = \ln(a) - \ln(b) \). Although not directly used in this exercise, it is a useful property to remember for other problems.

Understanding these properties allows us to transform expressions into simpler forms, making it easier to analyze limits and other mathematical concepts.

Convergent Sequences

Sequences are fundamental in understanding how functions behave as inputs grow large. A sequence is a set of numbers in a specific order, and a convergent sequence is one that approaches a specific value, called the limit, as the index approaches infinity.
The sequence given in the problem, \( a_n = \frac{2\ln(n)}{\ln(2) + \ln(n)} \), is an example of a convergent sequence. As \(n\) becomes very large, the contribution of \( \ln(2) \) in the denominator becomes less significant compared to \( \ln(n) \).
To determine whether a sequence is convergent, one can apply the concept of limits. If the sequence approaches a finite number as \(n\) tends to infinity, it converges. For this exercise, by simplifying and analyzing the behavior of \(a_n\) as \(n\) increases, we found that its limit is 2.
Recognizing and proving convergence involves analyzing how each term in the sequence behaves as it progresses towards infinity, often involving algebraic manipulations and limit theorems.

Infinite Limits

The concept of infinite limits helps us understand how functions or sequences behave as their input or index becomes exceedingly large. Unlike finite limits, where a function approaches a specific number, infinite limits describe the behavior of functions as they grow without bound.

• Limits at Infinity for Sequences: A sequence's limit at infinity is the value the sequence approaches as the index goes to infinity. For \(a_n\), the sequence converges, so it has a finite limit at infinity, which is 2.
• Dominance of Terms: For many functions, particularly those involving logarithms, understanding which terms "dominate" is crucial. In the given sequence, \( \ln(n) \) starts to dominate over \( \ln(2) \) in the denominator, simplifying \(a_n\) to 2, a constant value. This simplified analysis is key in solving limits at infinity.
• Determining Divergence or Convergence: When analyzing limits, it's important to note if they diverge (if they don't settle at a finite limit) or converge to a specific value. The mathematical simplifications help establish whether a sequence or function is convergent or divergent at infinity.

By using the concept of infinite limits, we gain insight into the long-term behavior of functions and sequences, crucial when dealing with growth and size analysis in mathematics.