Problem 21 - Thomas Calculus

Question

Determining Convergence or Divergence In Exercises \(17-44,\) use any method to determine if the series converges or diverges. Give reasons for your answer. $$\sum_{n=1}^{\infty} \frac{n !}{10^{n}}$$

Step-by-Step Solution

Answer

The series diverges by the Divergence Test.

Step 1: Identify the Series

The given series is \[ \sum_{n=1}^{\infty} \frac{n!}{10^n}. \] This is an infinite series where the general term is \( a_n = \frac{n!}{10^n} \). Our goal is to determine whether this series converges or diverges.

Step 2: Consider Divergence Test

Recall that the first test we can use is the Divergence Test. It states that if \( \lim_{{n \to \infty}} a_n eq 0 \), then the series diverges. Let us first apply this test.

Step 3: Compute the Limit of General Term

Evaluate the limit \( \lim_{{n \to \infty}} \frac{n!}{10^n} \). Since factorial \( n! \) grows faster than exponential \( 10^n \), \( n! \) will eventually dominate \( 10^n \). Thus \( \lim_{{n \to \infty}} \frac{n!}{10^n} = \infty \), which is non-zero.

Step 4: Conclusion from Divergence Test

Since the limit of the general term is not zero, by the Divergence Test, the series \( \sum_{n=1}^{\infty} \frac{n!}{10^n} \) diverges.

Key Concepts

Factorial Growth

Factorial growth is a fascinating mathematical concept that describes how rapidly the product of all positive integers up to a certain number grows. The factorial of a number, denoted by \(n!\), is the product of all positive integers less than or equal to \(n\). For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).
As \(n\) increases, \(n!\) grows extraordinarily fast, much faster than exponential and polynomial growth. This rapid increase is significant for studying series, as it influences convergence and divergence results.
In the original exercise, the factorial in the sequence \(\frac{n!}{10^n}\) means the terms grow very large as \(n\) becomes larger, which is a key factor in determining whether the series converges or not.

Divergence Test

The Divergence Test is a fundamental tool in analyzing whether an infinite series converges or diverges. It provides a quick way to identify divergent series. The test states: If the limit of the general term \(a_n\) of a series \(\lim_{{n \to \infty}} a_n eq 0\), then the series diverges.
This test, however, is not conclusive for proving convergence if the terms go to zero—the series might still diverge. It's crucial to use this test as a first check. If the terms do not vanish to zero, the series cannot converge.
In our exercise, the divergence test was used to find that \(\lim_{{n \to \infty}} \frac{n!}{10^n} = \infty\), thus confirming that the series \(\sum_{{n=1}}^{\infty} \frac{n!}{10^n}\) must diverge since the terms do not approach zero.

Infinite Series

An infinite series is the sum of the terms of an infinite sequence, usually represented as \(\sum_{n=1}^{\infty} a_n\). An infinite series can converge, meaning it approaches a specific value as we add more terms, or diverge, meaning it continues to grow without bound or oscillates indefinitely.
To determine convergence or divergence, various tests are applied, such as the divergence test, ratio test, and comparison test, among others. These tests help understand the behavior of the series.
In the exercise, the infinite series \(\sum_{n=1}^{\infty} \frac{n!}{10^n}\) was analyzed for convergence. By using the divergence test and observing the factorial growth, it was concluded that this particular series diverges, with terms continually increasing as \(n\) increases.