Question

Determine the convergence of the given series. State the test used; more than one test may be appropriate. $$\sum_{n=1}^{\infty} \frac{n !}{10^{n}}$$

Short answer

The series diverges by the Ratio Test.

Step-by-step solution

Understand the Series

We are given the series \( \sum_{n=1}^{\infty} \frac{n!}{10^n} \). Our goal is to determine if this series converges or diverges. This is a series composed of factorials, which grow very quickly.

Choose an Appropriate Test

For series involving factorials, the Ratio Test is generally effective because factorial growth can be easily compared to exponential terms like \(10^n\).

Apply the Ratio Test

The Ratio Test involves finding the limit of \(\left| \frac{a_{n+1}}{a_n} \right|\) as \(n \to \infty\). Here, \(a_n = \frac{n!}{10^n}\), and we compute:\[\frac{a_{n+1}}{a_n} = \frac{(n+1)! / 10^{n+1}}{n! / 10^n} = \frac{(n+1) \cdot 10^n}{10^{n+1}}\]\[= \frac{n+1}{10}\]

Evaluate the Limit

Now compute the limit:\[\lim_{n \to \infty} \frac{n+1}{10} = \lim_{n \to \infty} \left( \frac{n}{10} + \frac{1}{10} \right) = \infty\]Since the limit is \(\infty\), which is greater than 1, the series diverges according to the Ratio Test.

Conclude on Convergence

Given that the Ratio Test gives a result of \(\infty\) (greater than 1), the series \( \sum_{n=1}^{\infty} \frac{n!}{10^n} \) diverges. The rapidly increasing factorial \(n!\) overtakes the exponential growth of \(10^n\).

Key concepts

Ratio Test

The Ratio Test is a powerful tool for determining whether a series converges or diverges. This test is especially useful for series that involve factorials or exponential terms.
To apply the Ratio Test, you calculate 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 \( L = \infty \), the series diverges.- If \( L = 1 \), the test is inconclusive, and further analysis is needed.

The Ratio Test works well with the factorial involved in our given series because factorial terms grow quickly. In this case, when we applied the Ratio Test, the result was \( \infty \), indicating that the series diverges since the ratio exceeds 1.

Factorial Growth

Factorial growth is a concept reflecting how quickly numbers can increase when expressed in factorial form, denoted as \(n!\).
- A factorial of a non-negative integer \( n \) is the product of all positive integers less than or equal to \( n \).- For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).

Factorials grow at a staggering rate compared to polynomials or even exponential functions. This rapid growth makes it easy for factorials like \(n!\) to overpower other terms in a series. In the exercise, the factorial in the numerator \( n! \) exponentially increases, which contributes heavily to the divergence of the series. When compared to the exponential term in the denominator \(10^n\), the factorial outpaces it, leading to the conclusion that the series diverges. Understanding factorial growth is crucial when analyzing series with large terms.

Divergence of Series

The divergence of a series occurs when the sum of its terms does not approach a finite limit as the number of terms goes to infinity. If a series is divergent, its terms add up indefinitely rather than settling to a specific value.

There are several tests for checking divergence, and choosing the right one depends on the series form. In this example, the Ratio Test was chosen because it effectively handles the combination of factorial and exponential elements. The key takeaway is that if the test results in a value greater than 1 or approaches infinity, as we saw here, the series is confirmed to be divergent.

Divergence can signal that a series accumulates faster than expected, often indicating rapid growth of its terms like in our factorial scenario. Recognizing divergence helps in understanding the behavior of infinite series, which is fundamental in higher mathematics.