Problem 57 - Thomas Calculus

Question

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

Step-by-Step Solution

Answer

The series diverges by the ratio test.

Step 1: Understand the Series

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

Step 2: Apply the Ratio Test

The ratio test is useful for series with factorials. Consider \( a_n = \frac{(n !)^{n}}{(n^n)^2} \). The ratio test involves finding \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \).

Step 3: Simplify the Ratio

Compute \( a_{n+1} = \frac{((n+1)!)^{n+1}}{((n+1)^{n+1})^2} \). Hence, \( \frac{a_{n+1}}{a_n} = \frac{((n+1)!)^{n+1}}{((n+1)^{n+1})^2} \times \frac{(n^n)^2}{(n!)^n} \). Simplify this expression.

Step 4: Calculating the Limit

Factor and simplify \( \frac{((n+1)!)^{n+1}}{((n+1)^{n+1})^2} \times \frac{(n^n)^2}{(n!)^n} = \left( \frac{(n+1)!}{(n+1)^{n+1}} \right)^{n+1} \times \left( \frac{n^2}{n+1} \right)^n \). Apply Stirling's approximation if needed and take the limit as \( n \to \infty \).

Step 5: Determine Convergence or Divergence

After simplifying, if the limit is less than 1, the series converges. If equal to 1, the test is inconclusive. If greater than 1, the series diverges. In this case, the limit is \( \infty \), indicating divergence.

Key Concepts

Ratio Test

The ratio test is an analytical tool that helps determine whether certain infinite series converge or diverge. It is particularly handy when dealing with series containing factorials or exponential terms.

In essence, the ratio test examines the behavior of the terms in the series as the terms get closer to infinity. Here's how it works:

• Consider a series with terms \(\(a_n\)\).
• Compute the limit of the absolute value of the ratio between the \((n+1)^{th}\) term and the \(n^{th}\) term: \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\.
• Interpret the result:
• If the limit is less than 1, the series converges absolutely.
• If the limit is greater than 1, the series diverges.
• If the limit equals exactly 1, the test is inconclusive.

In our case, the series involves factorial expressions, making the ratio test an ideal choice to apply.

The test helps break down intricate factorial-based series into simpler comparisons, focusing on constituent term behaviors at infinity.

Stirling's Approximation

Stirling's approximation is a powerful mathematical tool used to simplify the computation of factorials, particularly for large numbers. It gives an approximate value that becomes increasingly accurate as the factorial's integer increases. The basic form of Stirling's approximation is:

• \[ n! \approx \sqrt{2 \pi n} \left( \frac{n}{e} \right)^n \]

This approximation helps to greatly simplify expressions by replacing the factorial with a more manageable form, especially when the series includes large factorials as in our problem.

By applying Stirling's approximation, we can transform the factorial terms into a form that is easier to analyze. This ultimately facilitates taking limits. When the approximation is used in the ratio test, it converts complex factorials into polynomials and exponentials that are easier to handle for calculations.

In the given exercise, using Stirling’s approximation is key to handling the term \(((n+1)!)^{n+1}\) effectively, making it feasible to analyze the behavior of the terms as \(n\) grows large.

Factorials

Factorials are a fundamental concept in mathematics, represented as \(n!\), where \(n\) is a non-negative integer. The factorial of a number is the product of all positive integers less than or equal to that number. It is denoted mathematically as:

• \[ n! = n \times (n-1) \times (n-2) \times \ldots \times 1 \]

Factorials grow very quickly with the increase of \(n\), making them a fitting element in sequences and series where rapid growth or degradation is under investigation.

Due to their rapid rate of growth, factorials can make series behave in unexpected ways. Hence, understanding how to manipulate them, either directly or through approximations like Stirling's, is crucial.

In the given exercise, \(n!\) appears in the function we are investigating in the series. Understanding how to manage these factorials is central to simplifying the terms and applying tests, like the ratio test, to decide on the convergence or divergence of the series.