Problem 37 - Thomas Calculus

Question

In Exercises \(17-46,\) use any method to determine if the series converges or diverges. Give reasons for your answer. $$ \sum_{n=1}^{\infty} \frac{n !}{(2 n+1) !} $$

Step-by-Step Solution

Answer

The series converges by the Ratio Test.

Step 1: Examine the Series Structure

The given series is \( \sum_{n=1}^{\infty} \frac{n!}{(2n+1)!} \). Here, the terms are of the form \( \frac{n!}{(2n+1)!} \), which suggests that we are dealing with factorials both in the numerator and the denominator.

Step 2: Use the Ratio Test

To determine whether the series converges or diverges, we will apply the Ratio Test. This involves computing the limit \( L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \), where \( a_n = \frac{n!}{(2n+1)!} \). So, \( a_{n+1} = \frac{(n+1)!}{(2(n+1)+1)!} = \frac{(n+1)!}{(2n+3)!} \).

Step 3: Simplify the Ratio

Now compute the ratio: \[ \frac{a_{n+1}}{a_n} = \frac{(n+1)!}{(2n+3)!} \times \frac{(2n+1)!}{n!}. \] Simplifying gives: \[ \frac{a_{n+1}}{a_n} = \frac{n+1}{(2n+3)(2n+2)(2n+1)}. \]

Step 4: Compute the Limit

Calculate the limit as \( n \to \infty \): \[ \lim_{n \to \infty} \frac{n+1}{(2n+3)(2n+2)(2n+1)} = \lim_{n \to \infty} \frac{1}{8n^2 + 12n + 3}. \] As \( n \to \infty \), this limit approaches 0.

Step 5: Conclude Based on the Ratio Test

Since the limit \( L = 0 < 1 \), the Ratio Test tells us that the series \( \sum_{n=1}^{\infty} \frac{n!}{(2n+1)!} \) converges.

Key Concepts

Factorial Series

A factorial series involves terms that are mathematical products called factorials. A factorial of a number \( n \), denoted as \( n! \), is the product of all positive integers less than or equal to \( n \). For instance, \( 4! = 4 \times 3 \times 2 \times 1 = 24 \).

Factorials grow rapidly as \( n \) increases. This fact makes them quite fascinating in infinite series like the one we are evaluating, \( \sum_{n=1}^{\infty} \frac{n!}{(2n+1)!} \). It's important to understand the nature of recent series, many of which feature factorial terms, given how quickly these terms can increase or decrease in size.

In our specific series, we find \( n! \) in the numerator and \( (2n+1)! \) in the denominator. This structure creates a sequence that diminishes because the denominator grows significantly faster than the numerator. Eventually, this leads to terms of the series shrinking towards zero, potentially making the series convergent.

Ratio Test

The Ratio Test is a powerful tool to determine convergence or divergence of an infinite series. It is particularly useful for series with factorials or exponential terms. Here's how it works:

• Identify \( a_n \), the nth term of the series.
• Calculate the limit \( L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \).
• If \( L < 1 \), the series converges. If \( L > 1 \), it diverges. If \( L = 1 \), the test is inconclusive.

In our series \( \sum_{n=1}^{\infty} \frac{n!}{(2n+1)!} \), we apply the Ratio Test by substituting \( a_n = \frac{n!}{(2n+1)!} \). By examining the ratio \( \frac{a_{n+1}}{a_n} = \frac{n+1}{(2n+3)(2n+2)(2n+1)} \), and taking the limit as \( n \to \infty \), we find that \( L = 0 \).

This result, \( L = 0 < 1 \), means that the series converges according to the Ratio Test.

Infinite Series

An infinite series is the sum of an infinite list of numbers, like \( a_1 + a_2 + a_3 + \ldots \). Mathematicians are often concerned with whether such a series converges (adds up to a finite number) or diverges (does not settle on a specific value).

Infinite series appear across many areas of mathematics and physics, forming the backbone of calculus concepts, particularly when analyzing continuous change or representing functions as sum of simple terms. Thus, tools and techniques to test series convergence or divergence, like the Ratio Test used here, are vital for deeper mathematical work.

In the case of our series \( \sum_{n=1}^{\infty} \frac{n!}{(2n+1)!} \), we concluded it converges by employing the Ratio Test, which shows how even a series with complex factorial components can settle to a finite sum.

Convergence and Divergence

Convergence and divergence are terms that describe whether an infinite series settles to a specific value or not as we add more terms. Understanding these concepts is crucial when dealing with series:

• Convergence: This occurs when the sum of an infinite series approaches a finite number.
• Divergence: This occurs when the sum either grows indefinitely or oscillates without approaching any specific value.

Applying these ideas to our series \( \sum_{n=1}^{\infty} \frac{n!}{(2n+1)!} \), the use of the Ratio Test showed convergence. Hence, the terms continuously shrink when summed indefinitely, leading to a series that limits to a finite value. This result highlights the intricacies involved in distinguishing between convergent and divergent series, an essential practice in mathematical studies.