Question

Determine whether the sequence converges or diverges. If convergent, give the limit of the sequence. $$\left\\{a_{n}\right\\}=\\{\ln (n)\\}$$

Short answer

The sequence \(\ln(n)\) diverges.

Step-by-step solution

Understanding the Problem

We need to determine whether the sequence \( a_n = \ln(n) \) converges or diverges. A sequence converges if it approaches a specific finite limit as \( n \to \infty \). Conversely, it diverges if it does not approach a particular finite value.

Analyzing the Sequence Behavior

Consider the sequence \( a_n = \ln(n) \). As \( n \to \infty \), the natural logarithm \( \ln(n) \) increases without bound. This means it does not settle on any finite value.

Concluding the Nature of the Sequence

Since \( \ln(n) \to \infty \) as \( n \to \infty \), the sequence \( a_n = \ln(n) \) diverges. A convergent sequence must approach a specific finite number, which this sequence does not.

Key concepts

Convergence

Convergence is a fundamental concept in the study of sequences and series. A sequence is said to converge if it approaches a particular finite value as the number of terms goes to infinity. In simpler terms, if you keep going further and further in the sequence and its terms get closer and closer to a specific number, then the sequence converges. This specific number is called the "limit" of the sequence. Here are a few important points about convergence:

• The sequence must settle on a single value.
• It implies stability as the number of terms increases.
• Not all sequences converge; some may diverge instead.

When a sequence converges, mathematicians are able to make precise predictions about the approximate value the sequence will approach, which is crucial in fields like calculus and analysis. In the problem given, the sequence in question does not meet the criteria for convergence because it does not approach any finite number.

Divergence

Divergence is the opposite of convergence. A sequence diverges when it does not approach any specific finite value as its number of terms increases. Instead, the terms of the sequence go off towards infinity or oscillate between various values without ever settling down. Divergence is an important concept because it helps us understand sequences that do not behave predictively.For a sequence to diverge:

• The terms increase or decrease without bound.
• The terms fluctuate in such a way that no single limit is approached.

In the original problem, the sequence \( a_n = \ln(n) \) demonstrates divergence. As \( n \to \infty \), the natural logarithm \( \ln(n) \) steadily increases. Since it does not approach a specific number, the sequence diverges. Divergent sequences are frequently found in various natural and social phenomena, where they often represent unbounded growth or unresolvable variability.

Natural Logarithm

The natural logarithm is a special mathematical function denoted by \( \ln(n) \). It is the inverse operation to exponentiation with the base "e," which is an irrational and transcendental constant approximately equal to 2.71828. Natural logarithms transform multiplication into addition, making them a powerful tool in mathematical calculations.Key features of natural logarithms include:

• \( \ln(1) = 0 \), because any non-zero number raised to the power of 0 equals 1.
• It is defined only for positive real numbers.
• \( \ln(e) = 1 \), by definition of the constant \( e \).
• The function \( \ln(n) \) increases without bound as \( n \) approaches infinity.

The natural logarithm is often used in settings that involve exponential growth or decay, such as biology, economics, and physics. In this particular exercise, the natural logarithm is integral to understanding why the sequence \( a_n = \ln(n) \) diverges.

Limit of a Sequence

The limit of a sequence is a fundamental concept that defines the behavior of sequences as their indices tend towards infinity. To determine whether a sequence converges or diverges, we look at the limit of the sequence. If the terms of the sequence approach a particular, finite number, then the limit exists and the sequence converges.Characteristics of a sequence limit include:

• The limit value is unique for convergent sequences.
• For divergence, the terms do not approach a single value.
• The precise definition involves approaching a number within any given small distance, no matter how small, provided the sequence terms are sufficiently far along.

In mathematical expression, if a sequence \( a_n \) has a limit \( L \), we say that \( \lim_{n \to \infty} a_n = L \). In this exercise, the sequence \( a_n = \ln(n) \) does not have a finite limit because it increases indefinitely. Understanding limits is essential in calculus and helps predict the long-term behavior of sequences.