Problem 25 - Calculus: Early Transcendentals

Question

Test the series for convergence or divergence. \( \displaystyle \sum_{n = 1}^{\infty} \frac {n!}{e^{n^2}} \)

Step-by-Step Solution

Answer

The series converges by the Ratio Test.

Step 1: Identify the Series

The given series is \( \sum_{n=1}^{\infty} \frac{n!}{e^{n^2}} \). We need to test this series for convergence or divergence.

Step 2: Apply the Ratio Test

The Ratio Test involves finding the limit of \( \left| \frac{a_{n+1}}{a_n} \right| \) as \( n \to \infty \), where \( a_n = \frac{n!}{e^{n^2}} \). First, find \( a_{n+1} = \frac{(n+1)!}{e^{(n+1)^2}} \).

Step 3: Compute the Ratio

Calculate \( \left| \frac{a_{n+1}}{a_n} \right| = \frac{(n+1)!}{e^{(n+1)^2}} \cdot \frac{e^{n^2}}{n!} = \frac{(n+1)}{e^{2n+1}} \).

Step 4: Find the Limit

Evaluate \( \lim_{n \to \infty} \frac{(n+1)}{e^{2n+1}} \). The exponential in the denominator grows much faster than the linear numerator, so the limit is 0.

Step 5: Apply the Conclusion from the Ratio Test

Since the limit \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = 0 < 1 \), by the Ratio Test, the series \( \sum_{n=1}^{\infty} \frac{n!}{e^{n^2}} \) converges.

Key Concepts

Ratio Test

The Ratio Test is a handy tool to determine whether an infinite series converges or diverges. For a series \( \sum a_n \), you calculate the ratio \( \left| \frac{a_{n+1}}{a_n} \right| \). If this limit as \( n \) approaches infinity exists and is less than 1, the series converges.
Conversely, if the limit is greater than 1, or if the limit does not exist, the series diverges. If the limit equals 1, the test is inconclusive.

• Calculate \( \left| \frac{a_{n+1}}{a_n} \right| \).
• Evaluate the limit as \( n \) approaches infinity.
• Apply the ratio criteria for convergence.

This test is crucial for series with factorials or exponential terms where simplicity and rapid growth might obscure convergence.

Factorial

The factorial of a number \( n \), written as \( n! \), is the product of all positive integers up to \( n \). So, \( n! = n \times (n-1) \times \ldots \times 1 \). Factorials grow very quickly as \( n \) increases, much faster compared to polynomial expressions.
For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). As illustrated here, the application of the Ratio Test often involves factorial expressions, making their understanding pivotal.

• Factorials are found in permutations and combinations.
• Recognizing their rapid growth is key in analyzing series with factorial terms.

Being familiar with factorials helps in calculating series convergence using tests like the Ratio Test in mathematical problems.

Exponential Growth

Exponential growth signifies that a quantity grows at a rate proportional to its current value. In mathematics, functions such as \( e^{n^2} \) represent exponential growth, where \( e \) (approximately 2.718) is the base of the natural logarithm.
Exponential terms increase very rapidly and often dominate other terms in expressions, especially for large \( n \).

• Essential in analyzing convergence where exponential terms are in denominators.
• Ubiquitous in natural processes and financial calculations.

By dominating factors like factorials in a series, the exponential component often indicates convergence due to its quicker growth rate as in our exercise where \( e^{n^2} \) offsets \( n! \).

Infinite Series

An infinite series is the sum of an infinite sequence of terms. We denote it as \( \sum_{n=1}^{\infty} a_n \) which symbolizes adding an endless count of sequence elements.
In calculating convergence of an infinite series, understanding how fast the terms \( a_n \) decrease is key. A series converges when its terms approach zero rapidly.

• Understanding behavior of the terms \( a_n \) leads to early detection of divergence or convergence.
• Methods like the Ratio Test or others help deduce convergence criteria.

In our exercise, understanding the interplay of \( n! \) over \( e^{n^2} \) within the sequence terms allows demonstrating convergence using the Ratio Test.