Problem 57 - Thomas Calculus

Question

Which of the series in Exercises \(57-64\) 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: Recognize the Series

The series we need to analyze is \( \sum_{n=1}^{\infty} \frac{2^{n} n! n!}{(2n)!} \). We aim to determine if this series converges or diverges.

Step 2: Apply the Ratio Test

The Ratio Test is useful for series involving factorials and exponential functions. Compute the limit:\[L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\] where\[ a_n = \frac{2^n n! n!}{(2n)!}.\]

Step 3: Calculate \(a_{n+1}\) and Form the Ratio

For \(a_{n+1}\), substitute \(n+1\) into the expression:\[a_{n+1} = \frac{2^{n+1} (n+1)! (n+1)!}{(2n + 2)!}.\]Now, form the ratio:\[\frac{a_{n+1}}{a_n} = \frac{\frac{2^{n+1} (n+1)! (n+1)!}{(2n + 2)!}}{\frac{2^n n! n!}{(2n)!}}.\]

Step 4: Simplify the Ratio

Simplify the ratio:\[\frac{a_{n+1}}{a_n} = \frac{2 \cdot (n+1)! (n+1)!}{(2n+2) (2n+1) \cdot n! n!}.\]This becomes:\[\frac{a_{n+1}}{a_n} = \frac{2 \cdot (n+1)(n+1)}{(2n+2)(2n+1)}.\]

Step 5: Evaluate the Limit

Evaluate the limit as \(n\) approaches infinity:\[L = \lim_{n\to\infty} \frac{2 \cdot (n+1)^2}{(2n+2)(2n+1)} = \lim_{n\to\infty} \frac{2(n^2 + 2n +1)}{4n^2 + 6n + 2}.\]Simplifying further gives:\[L = \lim_{n\to\infty} \frac{2n^2 + 4n + 2}{4n^2 + 6n + 2} = \frac{1}{2}.\]

Step 6: Conclude with the Ratio Test

Since \(L = \frac{1}{2} < 1\), by the Ratio Test, the series \(\sum_{n=1}^{\infty} \frac{2^n n! n!}{(2n)!}\) converges.

Key Concepts

Ratio Test

The Ratio Test is a vital tool in the evaluation of the convergence or divergence of an infinite series. It's especially useful when dealing with sequences containing factorials and exponential terms. To apply the Ratio Test, consider a series \(\sum_{n=1}^{\infty} a_n\). Calculate the ratio \(\left| \frac{a_{n+1}}{a_n} \right|\) and take the limit as \(n\) approaches infinity:\[ L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \]

• If \(L < 1\), the series converges absolutely.
• If \(L > 1\), the series diverges.
• If \(L = 1\), the test is inconclusive.

In our example, applying the Ratio Test to \(\sum_{n=1}^{\infty} \frac{2^n n! n!}{(2n)!}\) gives \(L = \frac{1}{2}\), which indicates convergence because \(\frac{1}{2} < 1\). This process is straightforward yet powerful for series with complex algebraic expressions.

Factorials in Series

Factorials often appear in infinite series, particularly those involving combinatorial expressions. When working with factorials, as in the series \(\sum_{n=1}^{\infty} \frac{2^n n! n!}{(2n)!}\), it’s essential to have a solid understanding of their properties.Factorially based terms in a series typically grow extremely fast. However, when they appear in both the numerator and denominator, as it happens in our example, they can sometimes balance each other and affect the convergence of the series.

• Recognition: Understanding the growth of \(n!\) is crucial. \(n!\) grows faster than any exponential sequence \(a^n\) for any constant \(a\).
• Simplification: Simplifying expressions involving factorials can clarify the overall behavior of the series. Consider how terms in the denominator reduce the sequence value, particularly when paired with similar or larger factorial expressions in the numerator.

This is evident in our given series, where the factorial growth in the numerator and denominator leads us to leverage the Ratio Test effectively.

Infinite Series Analysis

Analyzing infinite series is a process that allows one to understand and classify series based on their convergence properties. Infinite series can take numerous forms and require different strategies for inspection. The primary goal is to determine whether a series converges (approaches a finite limit) or diverges (does not settle to a finite limit). This evaluation relies on tests like the Ratio Test in the case of our example.Here are some general steps:

• Identification: Recognizing the structure of the series, whether it's geometric, harmonic, or involves factorials and exponentials helps in choosing the right test.
• Application of Tests: Using appropriate convergence tests such as the Ratio Test, the Root Test, or comparison tests based on the series components.
• Limit Evaluation: Accurately computing limits and series behavior as \(n\) approaches infinity.

In our exercise, the factorial and exponential terms prompted us to use the Ratio Test, which resulted in determining the series converges. Thorough understanding and application of these steps are essential for effective infinite series analysis.