Question

Use the ratio test to determine whether \(\sum_{n=1}^{\infty} a_{n}\) converges, where \(a_{n}\) is given in the following problems. State if the ratio test is inconclusive. $$ \sum_{n=1}^{\infty} \frac{\left(2^{n} n !\right)^{2}}{(2 n)^{2 n}} $$

Short answer

The ratio test is inconclusive since the limit equals 1.

Step-by-step solution

Identify Terms

Identify the general term of the series. Here, it is given by \(a_n = \frac{(2^n n!)^2}{(2n)^{2n}}\).

Find the Next Term

Calculate \(a_{n+1}\), which is the n+1 term of the series: \(a_{n+1} = \frac{(2^{n+1} (n+1)!)^2}{(2(n+1))^{2(n+1)}}\).

Form the Ratio

Form the ratio \(\frac{a_{n+1}}{a_n}\):\[\frac{a_{n+1}}{a_n} = \frac{\left(2^{n+1} (n+1)!\right)^2}{\left(2(n+1)\right)^{2(n+1)}} \times \frac{(2n)^{2n}}{(2^n n!)^2}\]

Simplify the Ratio

Simplify the expression for \(\frac{a_{n+1}}{a_n}\) step-by-step:1. Break down the factorial: \((n+1)! = (n+1) \cdot n!\).2. Split powers of 2: \(2^{n+1} = 2 \cdot 2^n\).3. Simplify the exponentials: Compute the terms \(\left(2(n+1)\right)^{2(n+1)}\) and \((2n)^{2n}\).4. Substitute back and simplify the resulting expressions.

Compute the Limit

Evaluate the limit \(\lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right|\). After simplification, this limit evaluates to 1.

Determine Convergence Using Ratio Test

According to the ratio test, if the limit \(L = \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right|\) is less than 1, the series converges; if greater than 1, it diverges; if equal to 1, the test is inconclusive. Here, \(L = 1\), thus the ratio test is inconclusive.

Key concepts

Convergence of Series

The concept of series convergence is pivotal in understanding whether an infinite series adds up to a finite number. A series is a sum of a sequence of terms that extends indefinitely. For a series to converge, the partial sums (sums of the first few terms) must approach a specific finite number as more terms are added.

Convergence is often determined using various tests, one of which is the Ratio Test. The ratio test checks the "growth" of terms in a series by comparing the ratio of successive terms, denoted as \( \frac{a_{n+1}}{a_n} \).

• If the limit of this ratio as \( n \to \infty \) is less than 1, the series converges absolutely.
• If the limit is greater than 1, the series diverges.
• However, if the limit equals 1, the test does nothing. This means other methods must assess convergence.

In this problem, the series resulted in a limit of 1, rendering the ratio test inconclusive. This requires exploring alternative convergence tests to conclude further.

Factorials in Series

Factorials play a crucial role in mathematics, especially within series and sequences. The factorial of a number \( n \), written as \( n! \), is the product of all positive integers up to \( n \). For example, \( 4! = 4 \times 3 \times 2 \times 1 = 24 \). This operation grows very quickly, influencing the behavior of series where factorials are present.

In the given series, factorials are part of the term: \( (n!)^2 \). When analyzing the convergence of series involving factorials, simplification becomes a skillful task.

• The factorial impacts how significantly the terms in the sequence grow. This rapid growth often necessitates the application of convergence tests, such as the Ratio Test, to understand the series's overall behavior.
• Simplifying factorial expressions is essential, such as interpreting \( (n+1)! = (n+1) \cdot n! \), to effectively analyze each term and their successive ratios.

In the exercise, recognizing factorial properties helped simplify the ratio of successive terms, enabling an evaluation of the series' convergence.

Limit Evaluation

Evaluating limits is a fundamental skill in calculus, often used to assess the behavior of functions or sequences as they approach infinity. A limit helps us understand how terms behave when extended indefinitely. In series analysis, determining the limit of the ratio of consecutive terms is a crucial step, particularly in Ratio Test applications.

In this exercise, the limit \[ \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\] was computed to determine convergence.

• Gather expressions and simplify to make limit evaluation manageable.
• Use properties of exponents and factorials to streamline calculations.
• Understand that the ratio \[ \frac{a_{n+1}}{a_n} \] represents the comparative growth of terms.

The limit evaluation in this case showed a result of 1. Such outcomes from the Ratio Test indicate that the test is inconclusive. Therefore, understanding how to compute and interpret limits is vital for series convergence analysis.