Problem 77 - Calculus: Early Transcendentals

Question

Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? \( a_n = 3 - 2ne^{-n} \)

Step-by-Step Solution

Answer

The sequence is increasing for \( n > 1 \) and bounded between 1 and 3.

Step 1: Define the Sequence & Terms

The given sequence is defined by the formula \( a_n = 3 - 2ne^{-n} \). We need to analyze this formula to determine whether the sequence is increasing, decreasing, or not monotonic as well as whether it is bounded.

Step 2: Examine Monotonicity with Derivative

To determine if the sequence is increasing or decreasing, analyze the derivative of the continuous analogue function \( f(x) = 3 - 2xe^{-x} \). Calculate the derivative and analyze its sign.\[ f'(x) = \frac{d}{dx}(3 - 2xe^{-x}) = 2x e^{-x} - 2e^{-x} = 2e^{-x}(x - 1) \]Since \( 2e^{-x} \) is always positive, the sign of \( f'(x) \) depends on \( (x - 1) \). Thus, the sequence is increasing when \( n > 1 \) and decreasing when \( n < 1 \).

Step 3: Evaluate the Nature at Critical Points

Evaluate the sequence values near \( n = 1 \) to determine the nature further. Check:- For \( n = 1 \), \( a_1 = 3 - 2e^{-1} \), which is a small decrement compared to 3.- For values as \( n \to \infty \), \( a_n \to 3 \).This shows fluctuations immediately around \( n = 1 \) but tends towards stabilization. Overall, the sequence tends monotonically to 3 after \( n = 1 \).

Step 4: Determine Boundedness

To check boundedness, consider that for all \( n \), the exponential term \( e^{-n} \) keeps \( 2ne^{-n} \) positive and constraine because \( e^{-n} \) approaches 0 as \( n \to \infty \). Thus, \( a_n = 3 - 2ne^{-n} \leq 3 \).Also, the lower bound can be observed since \( 2ne^{-n} \geq 0 \), ensuring \( a_n \geq 3 - 2ne^{-n} \approx 1 \) at bigger values of \( n \). Hence, the sequence is bounded between 1 and 3.

Key Concepts

Monotonicity

In calculus, a sequence is considered monotonic if it consistently moves in one direction, either increasing or decreasing, across its domain. To determine this for the sequence given by the formula \( a_n = 3 - 2ne^{-n} \), we use the concept of monotonicity. By evaluating the derivative of its continuous analogue \( f(x) = 3 - 2xe^{-x} \), we can deduce the nature of the sequence. The derivative is calculated as: \[ f'(x) = 2e^{-x}(x - 1) \] Here's what's important:

• When the expression \( x - 1 \) is positive (for \( x > 1 \)), \( f'(x) \) is positive, implying that the sequence is increasing.
• When \( x - 1 \) is negative (for \( x < 1 \)), \( f'(x) \) is negative, suggesting the sequence is decreasing.

Thus the sequence \( a_n \) is decreasing before \( n = 1 \) and increasing thereafter. This highlights that the sequence changes direction, showing a piecewise monotonic nature that eventually stabilizes.

Bounded Sequences

Bounded sequences are sequences that lie within a specific range, neither exceeding an upper bound nor dropping below a lower bound. In our case, to examine whether the sequence \( a_n = 3 - 2ne^{-n} \) is bounded, we first observe its behavior at large values of \( n \). As \( n \) increases, \( e^{-n} \) approaches zero, making the term \( 2ne^{-n} \) approach zero as well. Thus, \( a_n \) approximates 3. Key points include:

• The sequence never exceeds 3, affirming an upper bound.
• Observing \( a_n \), it also approximates values close to 1, establishing a lower bound.

Considering these bounds, \( a_n \) is a bounded sequence, specifically between 1 and 3. This bounded nature ensures that the sequence's values are restricted to this interval, contributing to its overall stability.

Calculus Problem Solving

Solving calculus problems often involves breaking down complex concepts into manageable steps. For sequences like \( a_n = 3 - 2ne^{-n} \), calculus helps analyze characteristics such as monotonicity and boundedness. Let's explore this process:

• Using Derivatives: Calculus allows us to find derivatives to understand the direction of a function. By differentiating \( f(x) = 3 - 2xe^{-x} \), we discover whether the sequence is increasing or decreasing.
• Evaluating Limits: Observing the behavior of \( a_n \) as \( n \) approaches infinity or zero can reveal the tendency of the sequence, aiding in spotting bounds.
• Analyzing Critical Points: Assessing specific values where changes occur, such as near \( n = 1 \), refines our understanding of sequences' dynamics.

Through these methods, calculus provides a solid foundation for dissecting the sequence, ensuring that each step brings us closer to understanding its properties and behavior fully.