Question

For each of the following sequences, whose \(n\) th terms are indicated, state whether the sequence is bounded and whether it is eventually monotone, increasing, or decreasing. $$ \ln \left(1+\frac{1}{n}\right) $$

Short answer

The sequence is bounded and eventually decreasing.

Step-by-step solution

Understanding the Sequence

The sequence given is \( \ln\left(1 + \frac{1}{n}\right) \) with \( n \in \mathbb{N}^+ \). This sequence represents the natural logarithm of \( 1 + \frac{1}{n} \), where \( n \) is a positive integer.

Determine Boundedness

To determine whether the sequence is bounded, note that for any positive \( n \), \( \frac{1}{n} \) is positive and decreasing towards zero. Therefore, \( 1 + \frac{1}{n} \) is always greater than 1, and as \( n \rightarrow \infty \), it approaches 1 from above. This results in \( \ln\left(1 + \frac{1}{n}\right) \to \ln(1) = 0 \). The sequence is bounded below by 0.

Verify Monotonicity

To examine whether the sequence is eventually monotone, consider the derivative of \( f(x) = \ln(1 + \frac{1}{x}) \). Differentiating gives \( f'(x) = -\frac{1}{x(x+1)} \), which is negative for all \( x > 0 \). This indicates that \( f(x) \) is decreasing for \( x > 0 \). Consequently, the sequence \( \ln\left(1 + \frac{1}{n}\right) \) is decreasing as \( n \) increases.

Key concepts

Monotonic Sequences

A monotonic sequence is a type of sequence where the terms consistently either do not decrease or do not increase. There are two primary types:

• Monotonically Increasing: Every term is greater than or equal to the term before it. This means that as you move forward in the sequence, the numbers either stay the same or get larger.
• Monotonically Decreasing: Every term is less than or equal to the term before it. This means that as you proceed with the sequence, the numbers either stay the same or get smaller.

In the case of the sequence given, which is specified by the term \( \ln\left(1 + \frac{1}{n}\right) \), we are interested in whether it is eventually monotonic. By examining its derivative, it was found that the sequence is decreasing. As \( n \) increases, the value of \( \frac{1}{n} \) decreases, leading to a smaller value inside the natural logarithm. This results in subsequent terms being smaller, confirming that the sequence is eventually monotonic and decreasing.

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is a fundamental mathematical function. It is the inverse of the exponential function, specifically the exponential with base \( e \), where \( e \approx 2.71828 \). The natural logarithm is defined only for positive real numbers.The natural logarithm has several key properties:

• The natural logarithm of 1 is 0: \( \ln(1) = 0 \).
• It is a strictly increasing function, meaning that if \( a < b \), then \( \ln(a) < \ln(b) \).
• It transforms products into sums, which is useful for simplifying multiplication: \( \ln(ab) = \ln(a) + \ln(b) \).

In our sequence \( \ln\left(1 + \frac{1}{n}\right) \), as \( n \rightarrow \infty \), \( 1 + \frac{1}{n} \) approaches 1. Thus, \( \ln\left(1 + \frac{1}{n}\right) \) steadily approaches the value of \( \ln(1) = 0 \). This shows how the natural logarithm interacts with this specific sequence, illustrating its behavior as the sequence progresses.

Limits of Sequences

The limit of a sequence is the value that the terms of the sequence approach as the index, typically denoted by \( n \), becomes very large. Understanding the limits of sequences is crucial in calculus and analysis as it helps in determining the behavior of sequences over time.To find the limit of a sequence \( a_n \), we look at what happens as \( n \rightarrow \infty \). A sequence \( a_n \) converges to the limit \( L \) if, as \( n \) increases without bound, the terms \( a_n \) get infinitely close to \( L \).In the given sequence \( \ln(1 + \frac{1}{n}) \), as \( n \to \infty \), the term \( \frac{1}{n} \to 0 \). This makes the expression inside the logarithm approach 1, and thus, \( \ln(1 + \frac{1}{n}) \to \ln (1) = 0 \). Therefore, the limit of this sequence is 0. Knowing limits is fundamental because it allows us to understand what value the sequence approaches over its progression, indicating that it is bounded and goes towards a specific number as \( n \) increases.