Question

Use the Squeeze Theorem to find the limit of each of the following sequences. $$ a_{n}=\frac{n !}{n^{n}} $$

Short answer

The limit of the sequence \( a_n = \frac{n!}{n^n} \) as \( n \to \infty \) is 0.

Step-by-step solution

Understand the Problem

We need to find the limit of the sequence \( a_n = \frac{n!}{n^n} \) as \( n \to \infty \). The Squeeze Theorem can be applied if we can find two sequences \( b_n \) and \( c_n \) such that \( b_n \leq a_n \leq c_n \) for all \( n \), and the limits of \( b_n \) and \( c_n \) are equal as \( n \to \infty \).

Set Upper and Lower Bounds

Recognize that \( 1 \leq n-i \leq n \) when \( i \leq n \). Therefore, \( i \leq n \) implies \( 1 \leq \frac{n-i}{n} < 1 \), which simplifies to \( 0 < a_n < \frac{1}{n} \).

Compute Bounds

For \( b_n \), set \( b_n = 0 \), as \( n! \) and \( n^n \) are positive, ensuring \( a_n > 0 \); and for \( c_n \), set \( c_n = \frac{1}{n} \), knowing \( a_n < \frac{1}{n} \).

Apply the Squeeze Theorem

Since \( 0 < a_n < \frac{1}{n} \) and we know \( \lim_{n\to\infty} 0 = 0 \) and \( \lim_{n\to\infty} \frac{1}{n} = 0 \), by the Squeeze Theorem, \( \lim_{n\to\infty} a_n = 0 \).

Conclusion

We have found two sequences, \( b_n = 0 \) and \( c_n = \frac{1}{n} \), that squeeze \( a_n \). Thus, by the Squeeze Theorem, \( \lim_{n\to\infty} a_n = 0 \).

Key concepts

Limits of Sequences

Understanding limits of sequences is vital in calculus and analysis. A sequence is a list of numbers that follow a particular order. When we analyze limits, we're interested in what happens to the terms of the sequence as the sequence progresses, especially as it approaches infinity.

To determine the limit of a sequence, you examine the behavior of its terms as the index (often denoted by \( n \)) becomes very large. This is expressed as \( n \to \infty \).

Sometimes, such as with the given sequence \( a_n = \frac{n!}{n^n} \), determining the limit is not straightforward. This is where tools like the Squeeze Theorem come in handy.

In simple terms, the Squeeze Theorem allows us to "squeeze" a sequence between two others whose limits are easier to calculate. If both bounding sequences converge to the same limit, then the original sequence also converges to that limit. This is highly useful in tackling sequences with complicated terms.

Factorial Growth

Factorial growth describes a situation in mathematics where the factorial function \( n! \) increases exceedingly fast compared to other mathematical functions. Factorials, represented by \( n! \), stand for the product of all positive integers up to a certain number \( n \). For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).

As \( n \) grows larger, \( n! \) speeds up tremendously. This rapid growth makes factorials a fascinating topic when studying limits of sequences and growth rates of functions.

In the sequence \( a_n = \frac{n!}{n^n} \), the numerator \( n! \) is growing factorially. However, the denominator \( n^n \) grows even faster because \( n^n \) means multiplying \( n \) by itself \( n \) times.

Factorial growth can seem overwhelming at first because it's less intuitive than linear or exponential growth. But understanding this concept helps in comprehending the behavior of sequences like the one in the exercise.

Asymptotic Analysis

Asymptotic analysis is a technique used to describe the limiting behavior of a function or sequence. It's particularly useful in understanding how sequences or functions behave as its parameters continue towards infinity.

When you perform an asymptotic analysis on a sequence, you're generally trying to find a simpler form of the sequence that describes its behavior as \( n \to \infty \). You're interested in the dominant terms or factors that dictate the growth or shrinkage of the function.

In the examined sequence \( a_n = \frac{n!}{n^n} \), asymptotic analysis focuses on how to simplify the complex factorial and power relationship. The faster growth of \( n^n \) relative to \( n! \) indicates that \( a_n \) shrinks rapidly as \( n \) increases, approaching zero.

By using asymptotic analysis, we can better understand why certain sequences grow, shrink, or stabilize, as well as leverage the useful mathematical theorems, like the aforementioned Squeeze Theorem, to find exact limits.