Problem 55 - Thomas Calculus

Question

Convergence or Divergence Which of the series in Exercises \(55-62\) converge, and which diverge? Give reasons for your answers. $$\sum_{n=1}^{\infty} \frac{2^{n} n ! n !}{(2 n) !}$$

Step-by-Step Solution

Answer

The series converges by the Ratio Test.

Step 1: Understand the Series

The given series is \( \sum_{n=1}^{\infty} \frac{2^n n! n!}{(2n)!} \). We need to identify whether this series converges or diverges. The general term is \( a_n = \frac{2^n n! n!}{(2n)!} \).

Step 2: Apply the Ratio Test

The Ratio Test helps determine convergence by examining the limit of \( \left| \frac{a_{n+1}}{a_n} \right| \) as \( n \) approaches infinity. So, \( a_{n+1} = \frac{2^{n+1} (n+1)! (n+1)!}{(2(n+1))!} \). We compute:\[\frac{a_{n+1}}{a_n} = \frac{2^{n+1} (n+1)! (n+1)!}{(2(n+1))!} \cdot \frac{(2n)!}{2^n n! n!}\]

Step 3: Simplify the Ratio

The expression simplifies to:\[\frac{a_{n+1}}{a_n} = \frac{2 \cdot (n+1)^2}{(2n+1)(2n+2)}\]Calculate the limit:\[\lim_{n \to \infty} \frac{2(n+1)^2}{(2n+1)(2n+2)} = \lim_{n \to \infty} \frac{2n^2 + 4n + 2}{4n^2 + 6n + 2} = \lim_{n \to \infty} \frac{2 + \frac{4}{n} + \frac{2}{n^2}}{4 + \frac{6}{n} + \frac{2}{n^2}} = \frac{1}{2}\]

Step 4: Conclusion from the Ratio Test

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

Key Concepts

Ratio Test

The Ratio Test is a powerful tool for determining the convergence of a series. It involves taking the limit of the absolute value of the ratio of consecutive terms of the series as the number of terms goes to infinity. Here's how it works:

• If this limit is less than 1, the series converges absolutely.
• If the limit is greater than 1, the series diverges.
• If the limit equals 1, the test is inconclusive.

To apply the Ratio Test effectively, consider a series with terms denoted by \( a_n \). The goal is to compute \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \). In our example, the terms involve factorials and exponents, which often simplify nicely when applying the Ratio Test, as they do here. You'll find that this approach elegantly resolves whether an infinite series converges or not by handling even complex terms with ease.
Continuing with the problem, once we compute the ratio for the given series and simplify it, we find that the limit is \( \frac{1}{2} \). Since \( \frac{1}{2} < 1 \), we conclude that the series converges.

Factorials in Series

Factorials play a pivotal role in many series problems, especially those involved in combinatorics and calculus. When working with factorial series, recognizing patterns in the manipulation of \( n! \) can simplify complex expressions. The factorial, \( n! \), is defined as the product of all positive integers up to \( n \). For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).
In the series in our exercise, we encounter terms like \( n! \) appearing in both the numerator and denominator. This allows us to cancel out several terms during simplification. By comparing consecutive terms, such as in the Ratio Test, the factorial terms often reduce significantly after simplifying, which makes it easier to evaluate the limit required for the test.

• Recognizing patterns and simplifications in factorial expressions is key to analyzing the series.
• Factorials grow rapidly, which often heavily influences the convergence behavior of a series.

Understanding how factorials interact within a series can make solving convergence problems much more manageable.

Infinite Series

An infinite series is an expression formed by adding an infinite number of terms: \( a_1 + a_2 + a_3 + \cdots \). To fully understand whether such a series converges or diverges, one must grasp a few key ideas and tests. A series converges if the sum of its terms approaches a specific finite number, as the series progresses to infinity. If it does not approach such a number, the series diverges.
Convergence indicates stability, meaning the sum does not keep growing without bound. Several tests can determine convergence, including:

• The Ratio Test (discussed above)
• The Root Test
• The Integral Test
• Comparison and Limit Comparison Tests

In the series from the exercise, checking for convergence is crucial in determining if the sum of its terms remains finite. Infinite series appear in many areas, such as physics, engineering, and financial mathematics, helping analyze phenomena ranging from wave patterns to the growth of investments over time. Each test, including our Ratio Test, gives insight into how the terms interact to form a convergent sum or reveal divergence.