Problem 10 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-20, an explicit formula for \(a_{n}\) is given. Write the first five terms of \(\left\\{a_{n}\right\\}\), determine whether the sequence converges or diverges, and, if it converges, find \(\lim _{n \rightarrow \infty} a_{n}\) \(a_{n}=e^{-n} \sin n\)

Step-by-Step Solution

Answer

The sequence converges to 0.

Step 1: Understanding the Sequence Formula

The given sequence is defined by the explicit formula \( a_n = e^{-n} \sin n \). This formula combines an exponential decay factor \( e^{-n} \) with the oscillating sine function \( \sin n \). The goal is to write out the first five terms of the sequence.

Step 2: Computing the First Five Terms

Calculate the first five terms of the sequence by substituting values of \( n = 1, 2, 3, 4, 5 \) into the formula:- \( a_1 = e^{-1} \sin(1) \)- \( a_2 = e^{-2} \sin(2) \)- \( a_3 = e^{-3} \sin(3) \)- \( a_4 = e^{-4} \sin(4) \)- \( a_5 = e^{-5} \sin(5) \)Find the approximate decimal values:- \( a_1 \approx 0.3092 \times 0.8415 \approx 0.2603 \)- \( a_2 \approx 0.1353 \times 0.9093 \approx 0.123 \)- \( a_3 \approx 0.0498 \times 0.1411 \approx 0.0070 \)- \( a_4 \approx 0.0183 \times -0.7568 \approx -0.0139 \)- \( a_5 \approx 0.0067 \times -0.9589 \approx -0.0064 \)

Step 3: Analyzing Convergence or Divergence

To determine whether the sequence converges or diverges, analyze the behavior of \( a_n = e^{-n} \sin n \) as \( n \to \infty \). The exponential term \( e^{-n} \) approaches zero as \( n \) increases, while \( \sin n \) continues to oscillate between -1 and 1.

Step 4: Conclusion on Convergence

Despite the oscillating nature of \( \sin n \), the product \( e^{-n} \sin n \) will approach zero because \( e^{-n} \) decays to zero faster than \( \sin n \) can vary. Thus, the sequence converges to zero.

Key Concepts

Explicit Formula

An explicit formula is a mathematical expression that allows us to find any term in a sequence directly. In our context, we have the explicit formula for the sequence \( a_n = e^{-n} \sin n \). This formula tells us that each term in the sequence is a product of two components:

• The exponential decay component \( e^{-n} \)
• The oscillating function component \( \sin n \)

Together, these dictate the behavior of each term in the sequence. To find, for instance, the second term \( a_2 \), simply substitute \( n = 2 \) into the formula. This method is simple, as it directly provides the value of any term without requiring previous terms. The ability to directly compute terms can be very useful for analyzing sequences, especially when determining if a sequence converges or diverges.

Exponential Decay

Exponential decay occurs when a quantity decreases at a rate proportional to its current value. In the sequence \( a_n = e^{-n} \sin n \), the \( e^{-n} \) term represents this decay. As \( n \) becomes larger, \( e^{-n} \) rapidly approaches zero. The reason is that the value of \( e^{-n} \) halves approximately every time \( n \) increases by one.
Here are some features of exponential decay:

• Rapid decrease: The term \( e^{-n} \) shrinks much faster than linear terms.
• Long-term behavior: As \( n \to \infty \), \( e^{-n} \to 0 \).

This aspect of the formula is critical when evaluating the convergence of the sequence. Even if the sine function causes oscillation, the rapid decay of the exponential term dominates, driving the values of the sequence towards zero.

Oscillating Function

The oscillating function \( \sin n \) in our sequence formula refers to its characteristic of moving up and down between -1 and 1 as \( n \) changes. This leads to a sequence with terms that vary between positive and negative values. Unlike exponential decay, the oscillation does not diminish as \( n \) increases. Here's what to know about oscillations in this context:

• Regular pattern: \( \sin n \) has a well-defined period of \( 2\pi \).
• Value range: It always stays within \(-1 \) to \( 1 \).

When combined with exponential decay, which trends towards zero, the oscillation is progressively less impactful on the long-term behavior of the sequence. This oscillating nature influences the initial terms but is eventually overtaken by the decay as \( n \) becomes large.

Limit of a Sequence

The limit of a sequence describes the value that the terms of a sequence approach as \( n \to \infty \). In mathematics, a sequence converges when its limit exists and is finite. For our sequence \( a_n = e^{-n} \sin n \), the limit evaluates whether the terms approach a specific value. We notice:

• As \( n \to \infty \), \( e^{-n} \) becomes very small, essentially negligible.
• No matter the value of \( \sin n \), the multiplying effect of \( e^{-n} \) ensures the terms approach zero.

Thus, the sequence converges to 0, because the rapid decay of \( e^{-n} \) dominates. The oscillating behavior of \( \sin n \) becomes inconsequential as \( n \) increases. Understanding this limit concept is crucial for fully grasping sequence behavior in calculus and analysis.