Question

Determine the limit of the sequence or show that the sequence diverges. If it converges, find its limit. $$ a_{n}=\ln \left(\frac{n+2}{n^{2}-3}\right) $$

Short answer

The sequence diverges to \(-\infty\).

Step-by-step solution

Identify the Form of the Sequence

The sequence given is \(a_n = \ln\left(\frac{n+2}{n^2-3}\right)\). It is a logarithmic function of a rational expression. Let's determine if it can be simplified for analysis of its limit.

Analyze the Expression Inside the Logarithm

Consider the expression inside the logarithm: \(\frac{n+2}{n^2-3}\). To find the behavior as \(n\) approaches infinity, analyze the highest power of \(n\) in the numerator and denominator. The numerator is \(n+2\), and the denominator is \(n^2-3\). The denominator grows much faster than the numerator.

Evaluate the Limit of the Expression

The highest degree term in the denominator \(n^2\) dominates, so \(\frac{n+2}{n^2-3} \approx \frac{n}{n^2} = \frac{1}{n}\). As \(n\to \infty\), \(\frac{1}{n} \to 0\). Hence, \(\lim_{{n \to \infty}} \frac{n+2}{n^2-3} = 0\).

Determine the Limit of the Logarithm

Since \(\lim_{{n \to \infty}} \frac{n+2}{n^2-3} = 0\), then \(\lim_{{n \to \infty}} a_n = \ln(0)\). However, \(\ln(0)\) is undefined. As the input to the logarithm approaches \(0\), \(\ln(x)\to -\infty\).

Conclude the Behavior of the Sequence

Since \(\ln(x)\) approaches \(-\infty\) as \(x\) approaches \(0\), the sequence \(a_n = \ln\left(\frac{n+2}{n^2-3}\right)\) diverges to \(-\infty\). We conclude that the sequence does not converge to any limit.

Key concepts

Logarithmic Function

A logarithmic function is a mathematical function that uses the logarithm as its main operation. The general form is \( \ln(x) \), where \( x \) must be a positive number. It describes how many times a base, often the natural base \( e \), must be multiplied by itself to reach a certain number. In practical terms, the log function measures the power to which the base is raised.
The specific sequence \( a_n = \ln\left(\frac{n+2}{n^2-3}\right) \) involves the natural logarithm. The logarithmic component helps explain the rate of growth or decay based on its input, which is a rational expression. Notably, logarithmic functions produce outputs that can be positive, negative, or undefined depending on the input value. Here, we use \( \ln(x) \) properties to explore sequence behavior as \( n \) becomes very large.

Rational Expressions

Rational expressions are fractions that involve polynomials. They include a numerator and a denominator, each of which can be linear or polynomial expressions. For the sequence given, \( \frac{n+2}{n^2-3} \) is a rational expression.
When analyzing rational expressions, we consider the highest degree terms because they dictate the behavior of the expression as \( n \) approaches infinity. Here, \( n+2 \) in the numerator simplifies to be dominated by \( n \) over the much larger\( n^2 \) in the denominator, leading to an approximation of \( \frac{1}{n} \) at infinity. Understanding this helps us determine that as \( n \to \infty \), \( \frac{1}{n^2} \) becomes insignificant compared to \( \frac{1}{n} \), thereby reducing the fraction's overall value towards zero.

Convergence and Divergence

Convergence refers to the behavior of a sequence where its terms approach a specific number or limit as \( n \) increases toward infinity. In contrast, divergence occurs when the sequence does not settle into a single value but instead grows without bound or oscillates.
In our sequence \( a_n = \ln\left(\frac{n+2}{n^2-3}\right) \), to determine convergence or divergence, we look at the limit of the sequence.

• If the limit of \( a_n \) exists and is finite, the sequence converges.
• If it leads to an undefined or infinite limit, like \( \ln(0) \), then the sequence diverges.

In this case, as \( n \to \infty \), the fraction \( \frac{n+2}{n^2-3} \) shrinks to zero. Thus, \( \ln(x) \to -\infty \) as \( x \to 0 \), showing the sequence diverges to \(-\infty\).

Behavior at Infinity

The behavior of a function or sequence as its input becomes very large, or approaches infinity, reveals significant long-term trends. These trends help in identifying whether the function approaches a particular value, zero, or tends towards infinity.
In our analysis, the rational expression \( \frac{n+2}{n^2-3} \) within the sequence dies down to zero due to the faster growth of the quadratic term \( n^2 \) in its denominator. As a result, the logarithmic function \( ln(x) \) becomes crucial. Since \( \ln(x) \) diverges negatively as \( x \) approaches zero, it implies that the sequence overall explodes towards negative infinity.

• The understanding of limits at infinity provides insights into how sequences behave over the long term.
• This helps identify if sequences like NaN (Not a Number), zeros, or infinities commence.

Here, "going to infinity" captures the diverging profile of \( a_n \), culminating in a result that clarifies its destiny towards \(-\infty\).