Problem 21 - Essential Calculus Early Transcendentals

Question

\(9-32\) n Determine whether the sequence converges or diverges. If it converges, find the limit. $$\left\\{\frac{(2 n-1) !}{(2 n+1) !}\right\\}$$

Step-by-Step Solution

Answer

The sequence converges to 0.

Step 1: Identify the Sequence Components

The given sequence is \(a_n = \frac{(2n-1)!}{(2n+1)!}\). This is a sequence of factorial expressions, and factorials grow rapidly. The components of the sequence are the factorial of odd numbers in the numerator and even numbers in the denominator.

Step 2: Simplify the Sequence

Rewrite the factorial in the denominator: \((2n+1)! = (2n+1) \cdot (2n) \cdot (2n-1)!\). Substitute this back into the sequence expression: \(a_n = \frac{(2n-1)!}{(2n+1)\cdot(2n)\cdot(2n-1)!}\).

Step 3: Cancel Out Common Terms

Notice that \((2n-1)!\) appears in both the numerator and the denominator, allowing us to cancel these terms out: \(a_n = \frac{1}{(2n+1) \cdot (2n)}\).

Step 4: Determine Behavior as n Approaches Infinity

Analyze the simplified sequence expression: \(a_n = \frac{1}{(2n+1)(2n)}\). As \(n\) approaches infinity, the denominator \((2n+1)(2n)\) increases without bound.

Step 5: Check Convergence

Since the denominator tends towards infinity, \(a_n\) approaches \(0\). Therefore, the sequence \(\left\{\frac{1}{(2n+1)(2n)}\right\}\) converges to 0.

Key Concepts

Factorial Sequence

Factorials are a key concept in mathematics and often appear in sequences. A factorial, represented by \(n!\), means multiplying a number by all positive integers below it. For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1\). Factorials grow really fast once you start increasing the numbers. In the context of sequences, a **factorial sequence** involves sequence terms that are expressed with factorials.

In the exercise, the sequence given is \(a_n = \frac{(2n-1)!}{(2n+1)!}\). This means both the numerator and the denominator include factorials of expressions. As factorials tend to become exceedingly large quickly, especially the ones in the denominator, sequences like these are highly influenced by how quickly the factorial in the denominator outweighs the one in the numerator. In this specific sequence, the factorial structure quickly helps determine the behavior of the sequence as the terms get larger and larger.

Limits

The concept of limits is vital in understanding sequence convergence. The **limit** of a sequence refers to the value that the sequence terms get closer to as they progress indefinitely. When solving sequences to determine if they converge or diverge, evaluating the limit is key.

For the exercise's sequence \(a_n = \frac{1}{(2n+1)(2n)}\), as \(n\) approaches infinity, both \(2n+1\) and \(2n\) grow without bound. This makes the denominator of the fraction become extremely large. Since the sequence term is a fraction where the numerator is 1 and the denominator increases continuously, the sequence's terms get closer to 0. Hence, for this sequence, the limit can be truly reached and is 0.

• When the limit of a sequence is finite, we can say the sequence converges to this limit.
• If the limit is infinite or doesn't settle at a number, the sequence diverges.

Infinite Sequences

Sequences can have a finite number of terms or extend indefinitely, which is when they are referred to as **infinite sequences**. These sequences keep going, and their behavior as they get larger is crucial in mathematics.

When evaluating infinite sequences, we are interested in their convergence or divergence. Convergence means the sequence approaches a specific number, while divergence implies it doesn't have a single limiting value. In the example given in the exercise, the sequence \(\frac{1}{(2n+1)(2n)}\) is an infinite sequence.

• By simplifying this sequence and analyzing its limit, we verified that as \(n\) went to infinity, the terms approached zero.
• This signifies convergence to the limit 0.
• Understanding infinite sequences is essential to calculus and advanced mathematics as they form the basis of many concepts and applications.

With infinite sequences, gently approaching the concept of convergence through simple examples is invaluable for students to grasp these abstract mathematical concepts.