Problem 20 - Thomas Calculus

Question

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 according to the Ratio Test.

Step 1: Identify the Series Type

The series given is \( \sum_{n=1}^{\infty} \frac{n !}{10^{n}} \). This is an infinite series and requires testing for convergence or divergence.

Step 2: Use the Ratio Test

To determine convergence, we'll use the Ratio Test. The Ratio Test is effective for series with factorials. For a series \( \sum a_n \), the ratio test involves computing the limit \( L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \). If \( L < 1 \), the series converges; if \( L > 1 \) or \( L \to \infty \), the series diverges; if \( L = 1 \), the test is inconclusive.

Step 3: Compute the Ratio

For our series, \( a_n = \frac{n !}{10^{n}} \). Compute:\[a_{n+1} = \frac{(n+1)!}{10^{n+1}} = \frac{(n!)(n+1)}{10^n \cdot 10}\]Now, find the ratio:\[\left| \frac{a_{n+1}}{a_n} \right| = \frac{(n!)(n+1)}{10^n \cdot 10} \div \frac{n!}{10^n} = \frac{(n+1)}{10}\]

Step 4: Evaluate the Limit

Calculate the limit of the ratio as \( n \to \infty \):\[L = \lim_{n \to \infty} \frac{n+1}{10} = \lim_{n \to \infty} \frac{n}{10} + \frac{1}{10} = \infty\]Since \( L = \infty \), which is greater than 1, the series diverges according to the Ratio Test.

Step 5: Conclusion

The series \( \sum_{n=1}^{\infty} \frac{n!}{10^n} \) diverges because the limit \( L \) from the Ratio Test is greater than 1, indicating divergence.

Key Concepts

Ratio Test

The Ratio Test is a powerful tool for determining the convergence or divergence of an infinite series. It's especially useful for series that include factorials. To use this test, you calculate the limit:

• Given a series \( \sum a_n \), compute \( L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \).
• If \( L < 1 \), the series converges.
• If \( L > 1 \) or \( L = \infty \), the series diverges.
• If \( L = 1 \), the test is inconclusive, and another method is necessary.

Once you identify the series, you determine the next term, calculate the ratio, and then find the limit as \( n \) approaches infinity. Understanding the Ratio Test helps in tackling more complex infinite series and aids in mastering series convergence topics.

Factorial in Series

The presence of a factorial in a series can drastically influence its behavior. Factorials grow extremely fast, often outpacing exponential functions in terms of rate of growth. This rapid escalation can significantly impact whether a series converges or diverges.
Factorials are represented with a '!' symbol, where \( n! = n \times (n-1) \times \ldots \times 2 \times 1 \). When evaluating a series like \( \frac{n!}{10^n} \), the factorial in the numerator increases sharply with each subsequent term.
For infinite series, including factorials typically necessitates a strong convergence test, such as the Ratio Test. Recognizing how factorials affect the growth of terms is crucial in predicting the behavior of series and understanding why certain series diverge rapidly.

Infinite Series Divergence

Infinite series can either converge to a specific value or diverge. Divergence means that the series does not settle to any specific number as you continue summing up its terms.
The divergence of a series like \( \sum_{n=1}^{\infty} \frac{n!}{10^n} \) can be determined through calculated testing methods such as the Ratio Test. When applying the Ratio Test to this series, we observe:

• The computed limit \( L = \infty \), which surpasses the threshold of 1.
• This indicates that the terms of the series grow too large and do not approach zero fast enough for the series to settle to a finite sum.

When \( L > 1 \), it conclusively shows that the series diverges. Understanding why a series diverges and mastering the application of convergence tests are vital skills in calculus, ensuring one can assess and predict the behavior of complex series.