Problem 16 - 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 \(\lim _{n \rightarrow \infty} a_{n}\). $$ a_{n}=\frac{n^{100}}{e^{n}} $$

Step-by-Step Solution

Answer

The sequence converges, and \( \lim_{n \to \infty} a_n = 0 \).

Step 1: Understanding the Formula

The given formula for the sequence is \( a_n = \frac{n^{100}}{e^n} \). This represents each term \( a_n \) in the sequence as the ratio of \( n^{100} \) to \( e^n \). Our task is to find the first five terms and analyze the behavior as \( n \) approaches infinity.

Step 2: Calculate First Five Terms

To calculate the first five terms, substitute \( n = 1, 2, 3, 4, \) and \( 5 \) into the formula.1. \( a_1 = \frac{1^{100}}{e^1} = \frac{1}{e} \)2. \( a_2 = \frac{2^{100}}{e^2} \)3. \( a_3 = \frac{3^{100}}{e^3} \)4. \( a_4 = \frac{4^{100}}{e^4} \)5. \( a_5 = \frac{5^{100}}{e^5} \) These are the first five terms of the sequence: \( \frac{1}{e}, \frac{2^{100}}{e^2}, \frac{3^{100}}{e^3}, \frac{4^{100}}{e^4}, \frac{5^{100}}{e^5} \).

Step 3: Analyze the Sequence for Convergence

We need to determine whether the sequence converges or diverges as \( n \to \infty \). The term \( e^n \) grows exponentially while \( n^{100} \) grows as a polynomial of degree 100. As \( n \to \infty \), the exponential term \( e^n \) will outgrow any polynomial, leading to \( a_n \to 0 \).

Step 4: Determine the Limit of the Sequence

Since the exponential denominator \( e^n \) grows faster than the polynomial numerator \( n^{100} \), we can conclude that the sequence converges to 0. Therefore, the limit is:\[ \lim_{n \to \infty} a_n = 0 \]

Key Concepts

Sequence Convergence

In mathematics, a sequence is said to converge if its terms approach a specific value as the term number approaches infinity. Understanding this concept is crucial in determining the behavior of infinite sequences.
When we talk about convergence, we're interested in seeing if the sequence settles down to a single, finite limit. In essence, we ask ourselves: "As we go further and further out in the sequence, do the numbers 'level off' to one particular number?" If they do, we say the sequence converges.
For example, in the provided exercise, the sequence given by \( a_n = \frac{n^{100}}{e^n} \) converges. As \( n \) increases, the exponential term \( e^n \) in the denominator dominates the growth of \( n^{100} \), causing the terms to shrink towards zero. Thus, we deduce that the sequence converges to 0.

• Convergence helps in estimating the sum of series derived from sequences.
• The limit of a converging sequence is the value that the terms get indefinitely close to as the sequence progresses.

Sequence Divergence

In contrast to convergence, a sequence is divergent if it does not approach a single finite limit. A divergent sequence might approach infinity or oscillate indefinitely without settling.
A basic understanding involves recognizing that if no single number captures the behavior of the entire sequence at infinity, it is divergent. Commonly, sequences that grow unbounded, either in positive or negative direction, perform this way.
For instance, if we considered a sequence \( b_n = n^2 \), then as \( n \) increases, \( b_n \) grows without bound—hence, it is divergent.

• Divergence indicates sequences that lack a pattern of 'settling down.'
• Some sequences can initially appear convergent but diverge as examined further.

Limits of Sequences

The limit of a sequence is a central theme when discussing convergence. The limit is the value that the terms of a sequence ``approach" as the term number becomes very large.
Knowing how to find limits helps us analyze the long-term behavior of sequences. For a sequence \( a_n \), the limit \( \lim_{n \to \infty} a_n \) is the value, if it exists, that \( a_n \) gets closer to as \( n \) becomes very large.
In our given exercise, we determined that the sequence's limit is 0, indicating that every term in the sequence tends towards zero as \( n \) increases. The execution of limits in sequences leverages some mathematical strategies:

• If the sequence numerator grows slower than the denominator, the sequence often converges to zero.
• If both parts of the sequence grow at equivalent rates, limits often require further investigation using techniques like L'Hôpital's Rule.

Grasping limits is fundamental for evaluating and understanding the broader behavior of sequences in calculus.