Question

Test the following series for convergence or divergence. Decide for yourself which test is easiest to use, but don't forget the preliminary test. Use the facts stated above when they apply. $$\sum_{n=0}^{\infty} \frac{(n !)^{2}}{(2 n) !}$$

Short answer

The series converges by the Ratio Test since \(L = \frac{1}{4} < 1\).

Step-by-step solution

- Recognize the Series

We have the series \(\sum_{n=0}^{\infty} \frac{(n !)^{2}}{(2 n) !}\). This is a series with terms involving factorials, so it is useful to consider the Ratio Test for convergence.

- Apply the Ratio Test

The Ratio Test states that for a series \(\sum a_n\) we consider 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 is infinite, the series diverges. If \(L = 1\), the test is inconclusive. Let's find \(L\) for our series.

- Compute \(a_{n+1}\)

First, we find the \((n+1)\)th term of the series: \(a_{n+1} = \frac{((n+1)!)^2}{(2(n+1))!} = \frac{(n+1)^2 \cdot (n!)^2}{(2n+2)!}\).

- Simplify \(a_{n+1}/a_n\)

Now we form the ratio \(\frac{a_{n+1}}{a_n}\): \(\frac{\frac{(n+1)^2 \cdot (n!)^2}{(2n+2)!}}{\frac{(n!)^2}{(2n)!}} = \frac{(n+1)^2 \cdot (n!)^2 \cdot (2n)!}{(2n+2)! \cdot (n!)^2} = \frac{(n+1)^2}{(2n+2)(2n+1)}\).

- Evaluate the Limit

Next, we take the limit as \ \ approaches infinity: \(L = \lim_{n \to \infty} \frac{(n+1)^2}{(2n+2)(2n+1)} = \lim_{n \to \infty} \frac{(n+1)^2}{4n^2 + 6n + 2}\). By dividing the numerator and the denominator by \^2\, we get \(L = \lim_{n \to \infty} \frac{1 + \frac{1}{n}^2}{4 + \frac{6}{n} + \frac{2}{n^2}}\). As \ approaches infinity, this expression simplifies to \(\frac{1}{4}\).

- Conclude Based on the Ratio Test

Since \(L = \frac{1}{4} < 1\), by the Ratio Test, the series \(\sum_{n=0}^{\infty} \frac{(n !)^{2}}{(2 n) !}\) converges.

Key concepts

Ratio Test

The Ratio Test is a tool used to determine the convergence or divergence of an infinite series. For a series \(\textstyle\sum a_n \), we consider the limit \(\textstyle L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \).
The Ratio Test provides the following criteria for convergence:

• If \(\textstyle L < 1 \), the series converges.
• If \(\textstyle L > 1 \) or is infinite, the series diverges.
• If \(\textstyle L = 1 \), the test is inconclusive and other methods must be used.

This test is particularly useful for series involving factorials, exponential functions, or terms with complex multiplicative components. In our exercise, the series \(\textstyle\sum_{n=0}^{\infty} \frac{(n !)^{2}}{(2 n) !} \), contains factorials, making the Ratio Test an optimal choice for determining its convergence.

Factorials

Factorials, denoted as \(\textstyle n! \), represent the product of all positive integers up to \(\textstyle n \). For example, \(\textstyle 5! = 5 \times 4 \times 3 \times 2 \times 1 \).
Factorials grow very quickly with increasing \(\textstyle n \), a property that significantly influences the behavior of series involving factorials.
In the given series \(\textstyle \sum_{n=0}^{\infty} \frac{(n !)^{2}}{(2 n) !} \), the presence of factorials in both the numerator and the denominator affects the rate of convergence. Understanding how factorials expand and simplify is essential when performing manipulations, such as those in the Ratio Test.

Infinite Series

An infinite series is the sum of an infinite sequence of terms. Infinite series can either converge to a finite value or diverge. Convergence means the sum approaches a particular number as more terms are added. Divergence means the sum either increases without bound or oscillates indefinitely without settling down.
For our series \(\textstyle\sum_{n=0}^{\infty} \frac{(n !)^{2}}{(2 n) !} \), the task is to determine if it converges (adds up to a finite value) or diverges using various methods, including the Ratio Test.

Limit Evaluation

Limit evaluation is a critical technique in calculus used to find the value a function approaches as the input approaches a given point. When applying the Ratio Test, after forming the ratio \(\textstyle\frac{a_{n+1}}{a_n} \), we need to evaluate the limit \(\textstyle\lim_{n \to \infty} \frac{a_{n+1}}{a_n} \).
In our example, this process involves simplifying the expression: \(\textstyle\lim_{n \to \infty} \frac{(n+1)^2}{4n^2 + 6n + 2} \).
By dividing the numerator and the denominator by \(\textstyle n^2 \), we simplify to: \(\textstyle\lim_{n \to \infty} \frac{1 + \frac{1}{n^2}}{4 + \frac{6}{n} + \frac{2}{n^2}} \), which approaches \(\textstyle\frac{1}{4} \) as \(\textstyle n \) goes to infinity. This value is less than 1, indicating that the series converges.

Limit Evaluation

Limit evaluation is a critical technique in calculus used to find the value a function approaches as the input approaches a given point. When applying the Ratio Test, after forming the ratio \(\textstyle\frac{a_{n+1}}{a_n} \), we need to evaluate the limit \(\textstyle\lim_{n \to \infty} \frac{a_{n+1}}{a_n} \).
In our example, this process involves simplifying the expression: \(\textstyle\lim_{n \to \infty} \frac{(n+1)^2}{4n^2 + 6n + 2} \).
By dividing the numerator and the denominator by \(\textstyle n^2 \), we simplify to: \(\textstyle\lim_{n \to \infty} \frac{1 + \frac{1}{n^2}}{4 + \frac{6}{n} + \frac{2}{n^2}} \), which approaches \(\textstyle\frac{1}{4} \) as \(\textstyle n \) goes to infinity. This value is less than 1, indicating that the series converges.