Problem 10 - Calculus Early Transcendentals: Pearson New International Edition

Question

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 and \(\lim_{n \to \infty} a_n = 0\).

Step 1: Calculate the First Term

To find the first term, substitute \(n = 1\) into the explicit formula \(a_n = e^{-n} \sin n\). We calculate: \(a_1 = e^{-1} \sin 1\).

Step 2: Calculate the Second Term

To find the second term, substitute \(n = 2\) into the formula: \(a_2 = e^{-2} \sin 2\).

Step 3: Calculate the Third Term

To find the third term, substitute \(n = 3\): \(a_3 = e^{-3} \sin 3\).

Step 4: Calculate the Fourth Term

For the fourth term, substitute \(n = 4\): \(a_4 = e^{-4} \sin 4\).

Step 5: Calculate the Fifth Term

For the fifth term, substitute \(n = 5\): \(a_5 = e^{-5} \sin 5\).

Step 6: Analyze Convergence

Observe that as \(n\) approaches infinity, the term \(e^{-n}\) approaches 0 because the exponential function diminishes rapidly. Therefore, \(e^{-n} \sin n\) will also tend towards 0, as \(e^{-n}\) dictates the behavior of the sequence.

Step 7: Calculate the Limit

Since \(e^{-n}\) approaches 0 and \(\sin n\) oscillates between -1 and 1, the sequence \(a_n = e^{-n} \sin n\) tends to 0 as \(n\) approaches infinity. Thus, \(\lim_{n \to \infty} a_n = 0\).

Key Concepts

Explicit Formula

An explicit formula provides a direct way to calculate any term in a sequence without relying on the previous terms. It's like having a recipe that tells you how to cook any dish directly. For the sequence given, the explicit formula is \(a_n = e^{-n} \sin n\). This formula allows us to find each term without needing to know the previous one.
To use this formula effectively:

• Substitute the desired value for \(n\) into the formula.
• Calculate the exponential part \(e^{-n}\), which decreases rapidly as \(n\) increases.
• Calculate the sine function \(\sin n\) which oscillates between -1 and 1.

By directly plugging in the values, as shown in the original solution, we can determine each term in the sequence, like \(a_1 = e^{-1} \sin 1\), \(a_2 = e^{-2} \sin 2\), and so on.

Limit of a Sequence

The limit of a sequence refers to the value that the terms of a sequence approach as \(n\) becomes very large. In other words, it's what the sequence "settles into" as you go far out.To determine if a sequence converges to a limit:

• Observe the behavior of the sequence's formula as \(n\) approaches infinity.
• If the terms get closer to a single value, the sequence converges. Otherwise, it diverges.

For the sequence \(a_n = e^{-n} \sin n\):

• The term \(e^{-n}\) approaches 0 because exponential functions decrease rapidly to 0.
• The function \(\sin n\) remains bounded between -1 and 1, but its influence diminishes as \(e^{-n}\) becomes very small.
• Therefore, the sequence converges to 0.

This is mathematically expressed as \(\lim_{n \to \infty} a_n = 0\), indicating that the sequence stabilizes at 0.

Oscillating Function

An oscillating function moves back and forth in a regular pattern, like a pendulum. In mathematics, functions like \(\sin x\) naturally oscillate because they repeatedly swing between set bounds, usually between -1 and 1.Key characteristics of an oscillating function include:

• They do **not settle** into a single value.
• Their pattern is often periodic, repeating after certain intervals.
• Despite oscillating, they can pair with other functions to result in convergence scenarios.

In the formula \(a_n = e^{-n} \sin n\):

• \(\sin n\) is the oscillating element, swinging between -1 and 1.
• When multiplied by \(e^{-n}\), which tends to 0, the overall term becomes vanishingly small, resulting in convergence to 0.

Thus, despite the oscillating nature of \(\sin n\), the sequence's overall behavior is dominated by \(e^{-n}\), illustrating how damping effects work in mathematical sequences.