Problem 41 - Calculus: Early Transcendentals

Question

Determine whether the sequence converges or diverges. If it converges, find the limit. \( \\{ n^2e^{-n}\\} \)

Step-by-Step Solution

Answer

The sequence converges to 0.

Step 1: Understand the Sequence

The sequence in question is \( \{ n^2e^{-n} \} \). This sequence has the general term \( a_n = n^2e^{-n} \).

Step 2: Simplify Expression

To analyze this sequence, rewrite it using the exponential property. Note that \( e^{-n} = \frac{1}{e^n} \). Hence, the term becomes \( a_n = \frac{n^2}{e^n} \).

Step 3: Apply Limits

Consider the behavior as \( n \to \infty \). The exponential term \( e^n \) grows much faster than the polynomial term \( n^2 \), so \( a_n = \frac{n^2}{e^n} \) tends towards 0.

Step 4: Use L'Hôpital's Rule if Needed

L'Hôpital's Rule can confirm this: for indeterminate forms like \( \frac{\infty}{\infty} \), differentiate numerator and denominator. For \( \frac{n^2}{e^n} \), \( \frac{d}{dn}(n^2) = 2n \) and \( \frac{d}{dn}(e^n) = e^n \). The limit becomes \( \lim_{n \to \infty} \frac{2n}{e^n} \), still \( \frac{\infty}{\infty}\). Apply L'Hôpital's again: \( \frac{d}{dn}(2n) = 2 \) and \( \frac{d}{dn}(e^n) = e^n \). The limit is \( \lim_{n \to \infty} \frac{2}{e^n} = 0 \).

Step 5: Conclude on Convergence

Since \( \lim_{n \to \infty} a_n = 0 \), the sequence \( \{ n^2e^{-n} \} \) converges to 0.

Key Concepts

L'Hôpital's Rule

When faced with complex limits, like fractions where both the numerator and the denominator approach infinity, we can use L'Hôpital's Rule. This powerful technique simplifies evaluating such indeterminate forms, like \( \frac{\infty}{\infty} \).L'Hôpital's Rule states that if you have an indeterminate form, you can differentiate the numerator and denominator separately until determining the limit becomes straightforward.Here's a simple process:

• Identify if the limit is an indeterminate form, often \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \).
• Differentiating the numerator and the denominator separately.
• Evaluate the limit again to see if the form changes to a determinable form. Repeat the differentiation process if necessary.

Applying this to our example \( \frac{n^2}{e^n} \), we differentiate to get \( \frac{2n}{e^n} \) and then \( \frac{2}{e^n} \) on subsequent steps.Eventually, as \( e^n \) grows exponentially, the fraction approaches 0. This confirms the sequence converges. Remember, L'Hôpital's Rule is handy for complex limits, ensuring we don't get stuck with "undefined" loops.

Exponential Growth

Exponential growth describes how a quantity increases rapidly, doubling or more at consistent intervals. This concept is pivotal in understanding why certain mathematical expressions outweigh others as they increase.The exponential function, \( e^x \), exemplifies exponential growth. As \( x \) increases, \( e^x \) increases very fast. Consider a sequence like \( n^2e^{-n} \), rewritten as \( \frac{n^2}{e^n} \). Here, \( e^n \) in the denominator grows rapidly, much faster compared to the \( n^2 \) in the numerator.A few points about exponential growth:

• It's faster than any polynomial growth like \( n^2 \) or \( n^3 \).
• Even small exponential terms (like \( e^{-n} \)) decrease quickly to 0 as \( n \) becomes large.
• This rapid growth is why, as \( n \to \infty \), \( \frac{n^2}{e^n} \) becomes essentially 0.

Remember, exponential terms dominate polynomial ones, significantly impacting sequences and their convergence.

Limit of a Sequence

Limits help us understand the behavior of sequences as they extend towards infinity. If a sequence converges, it settles towards a number called the limit.For the sequence \( \{n^2e^{-n}\} \), rewritten as \( \frac{n^2}{e^n} \), we observed its limit as \( n \) approaches infinity. Here, the exponential \( e^n \) grows faster than \( n^2 \), driving the limit towards zero.Here's how to think about sequence limits:

• The idea is to see if, as \( n \) grows, the terms of the sequence approach a single value.
• If the values stabilize around a number, like 0 here, the sequence converges to that limit.
• If they don't settle, the sequence diverges, meaning no limit exists at infinity.

The analysis of limits is foundational in calculus, helping predict sequence behavior for extremely large terms. Understanding these key aspects empowers students to tackle related problems and see the broader applications beyond pure math.