Problem 18 - Calculus: Early Transcendentals

Question

Use the Ratio Test to determine whether the series is convergent or divergent. \( \displaystyle \sum_{n = 1}^{\infty} \frac {n!}{n^n} \)

Step-by-Step Solution

Answer

The series converges.

Step 1: Write Down the Terms of the Series

The given series is \( \sum_{n=1}^{\infty} \frac{n!}{n^n} \). Identify that the general term of the series is \( a_n = \frac{n!}{n^n} \).

Step 2: Apply the Ratio Test Formula

To apply the Ratio Test, evaluate \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \). For our series, this means evaluating the limit: \[ \lim_{n \to \infty} \left| \frac{\frac{(n+1)!}{(n+1)^{n+1}}}{\frac{n!}{n^n}} \right|. \]

Step 3: Simplify the Expression Within the Limit

Simplify the expression for the ratio: \[ \frac{a_{n+1}}{a_n} = \frac{(n+1)!}{(n+1)^{n+1}} \times \frac{n^n}{n!}. \] This simplifies further to: \[ \frac{(n+1) \cdot n^n}{(n+1)^{n+1}}. \]

Step 4: Simplify by Canceling Terms

Continue simplifying: \[ \frac{a_{n+1}}{a_n} = \frac{n^n}{(n+1)^n}. \] This is because \((n+1)\) cancels out one power of \(n+1\) in the denominator.

Step 5: Evaluate the Limit

Now evaluate the limit: \[ \lim_{n \to \infty} \frac{n^n}{(n+1)^n} = \lim_{n \to \infty} \left( \frac{n}{n+1} \right)^n. \] Notice this expression is \( \left( 1 - \frac{1}{n+1} \right)^n \).

Step 6: Recognize the Form and Conclude

As \( n \to \infty \), the expression \( \left( 1 - \frac{1}{n+1} \right)^n \) is equivalent to \( \frac{1}{e} \) due to the exponential limit property. Since \( \frac{1}{e} < 1 \), the series converges according to the Ratio Test.

Key Concepts

Series Convergence

When discussing whether a series converges or diverges, we refer to how the sum of its infinite terms behaves. If the series converges, the sum approaches a specific finite number. However, if it diverges, the sum grows indefinitely or oscillates without settling down.
To determine convergence, we can use various tests, with the Ratio Test being one of the most effective methods for series involving factorials or powers. In the context of this exercise, we applied the Ratio Test to the series \( \sum_{n=1}^{\infty} \frac{n!}{n^n} \), revealing it converges.
Understanding the nature of each term in the series is crucial. For our series, each term \( a_n = \frac{n!}{n^n} \) contributes to whether the sum will ultimately converge.

Factorial

Factorial is a crucial concept in the evaluation of series involving factorial terms. Denoted as \( n! \), the factorial of a non-negative integer \( n \) is the product of all positive integers up to \( n \).
It can grow very rapidly, which often influences convergence when it appears in a series. In our given series, it plays a key role in the term \( \frac{n!}{n^n} \).

• \( 0! = 1 \)
• \( 1! = 1 \)
• Increasing factorials: \( n! = 1 \cdot 2 \cdot 3 \cdots n \)

Though factorials can result in very large values, the factorial term in our series is balanced by the denominator \( n^n \), influencing the series' overall behavior.

Limit Evaluation

Evaluating limits is an art when determining series convergence, especially through tests like the Ratio Test. We focus on the limit of the ratio of successive terms in the series.
For this exercise, we look at:\[\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n \to \infty} \frac{n^n}{(n+1)^n}.\]Breaking this down helps us determine convergence. Simplifying \( \left(\frac{n}{n+1}\right)^n \) gives insights due to the known limit behaviors.
Evaluating such limits is pivotal—here, it guides us to conclude convergence within the Ratio Test by revealing a well-known exponential limit that we explore next.

Exponential Limit Property

The exponential limit property becomes crucial here when evaluating expressions of the form:\[\left( 1 - \frac{1}{n+1} \right)^n.\]Such expressions often appear in convergence problems and relate closely to the exponential number \( e \). As \( n \) increases to infinity, the expression converges to \( \frac{1}{e} \).

• \( e \approx 2.71828 \) is the base of natural logarithms.
• Familiar limits: \( \lim_{n \to \infty} \left(1 - \frac{x}{n}\right)^n = e^{-x} \)
• In our example, substituting \( x = 1 \) gives \( \frac{1}{e} \)

Recognizing these patterns and properties is essential. In this problem, concluding that the limit equals \( \frac{1}{e} \) confirms that the series converges since \( \frac{1}{e} < 1 \). This type of limit evaluation is crucial when employing the Ratio Test for series convergence.