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. $$ n^{-1 / n}, n \geq 3 $$

Short answer

The sequence is bounded and eventually decreases.

Step-by-step solution

Analyze the Boundedness of the Sequence

The sequence is given by \( a_n = n^{-1/n} \). To check if the sequence is bounded, we need to observe the values of \( a_n \) as \( n \) approaches infinity. Notice that \( a_n \) are positive, so the lower bound is 0. As \( n \to \infty \), \( n^{-1/n} = e^{-(rac{1}{n}) \ln(n)} \) decreases as \( n \) increases, leading to \( n^{-1/n} \to 1^+=1 \). So, the sequence is bounded within (0, 1].

Determine Monotonicity of the Sequence

To determine if \( a_n = n^{-1/n} \) is eventually increasing or decreasing, examine the derivative. Let \( f(n) = n^{-1/n} \), then \( \ln(f(n)) = -\frac{1}{n} \ln(n) \). Differentiating with respect to \( n \) gives \( f'(n) = f(n) \left( \frac{\ln(n) + 1}{n^2} \right) \). For \( n > e \), this derivative simplifies to a negative value indicating that \( a_n \) is decreasing for all \( n \geq 3 \).

Conclusion About Sequence

Summarizing the above analysis: The sequence \( n^{-1/n} \) is bounded and eventually decreases as \( n \geq 3 \). This fits the characteristics of a bounded decreasing sequence in the interval (0, 1].

Key concepts

Bounded Sequences

A bounded sequence in mathematics means that the elements of the sequence are confined within a specific range. When examining sequences, it's important to check if they have upper and lower limits. For instance, if a sequence is bounded, there exists a number that no term in the sequence exceeds, known as the upper bound, and another number that every term in the sequence is greater than, known as the lower bound.
In the given example, the sequence is determined by the formula \( a_n = n^{-1/n} \). It is bounded because each term \( a_n \) lies between 0 and 1. This means as \( n \) becomes very large, the sequence remains confined within this interval (0, 1], providing stability and predictability in its behavior.

• The lower bound here is 0 because \( a_n \) is always positive.
• The upper bound is 1 since \( n^{-1/n} \) approaches 1 as \( n \) grows.

This bounded nature is essential for understanding how sequences behave over the long term and for determining whether they converge to a particular value.

Monotonic Sequences

A sequence is termed monotonic if it consistently either increases or decreases. The concept of monotonicity helps identify the general trend in the sequence's behavior. A sequence that increases steadily is known as increasing, while one that decreases is known as decreasing.
Analyzing the sequence \( a_n = n^{-1/n} \), we find that it is not entirely monotonic but rather eventually becomes monotonic. Initially, a sequence might not follow a monotonic pattern, but after a certain number of terms, it can settle into an increasing or decreasing trend.
This sequence turns out to be decreasing for \( n \geq 3 \), as shown by the derivative analysis in the solution, indicating that the values of \( a_n \) get smaller as \( n \) increases further. Therefore, while not strictly decreasing from the start, \( a_n \) eventually falls into this predictable pattern, classifying it as an eventually monotonic sequence.

Increasing and Decreasing Sequences

An important distinction within monotonic sequences is whether they are specifically increasing or decreasing. An increasing sequence is one where each term is equal to or greater than the preceding term. Conversely, a decreasing sequence is defined as one where each term is equal to or less than the preceding term.
The sequence \( a_n = n^{-1/n} \) as demonstrated, becomes a decreasing sequence when \( n \geq 3 \). This means every term is smaller than or similar to the one before it, forming a descending series as \( n \) grows.

• For decreasing sequences, the derivative of the term is usually negative, as in this example.
• For \( a_n = n^{-1/n} \), after computation, \( f'(n) \) was shown to be negative, confirming this decreasing pattern.

This characteristic is crucial in calculus as it aids in predicting the sequence's behavior over extended intervals. Also, understanding these patterns allows mathematicians to analyze convergence and limit behavior accurately.

Limits of Sequences

The concept of a sequence's limit is central to calculus and analysis. A sequence has a limit if its terms approach a specific value as \( n \) becomes infinitely large. If such a value exists, it tells us what the sequence tends to after a prolonged series of terms.
In the case of \( a_n = n^{-1/n} \), as discussed in the solution, the sequence approaches and converges to the limit 1 as \( n \) increases. This happens because the expression \( e^{-(\frac{1}{n}) \ln(n)} \) nears the value 1, demonstrating a specific target value, showing the bounded and decreasing nature of the sequence.
Recognizing the limit of a sequence:

• Provides insights into the eventual value the sequence will approximate.
• Confirms the predictability of long-term behavior.

This satisfying limit also reinforces the stability and bounded characteristics of the sequence, ensuring it never diverges or moves towards infinity.