Problem 7 - Calculus: Early Transcendentals

Question

List the first five terms of the sequence. \( a_n = \frac {1}{n + 1}! \)

Step-by-Step Solution

Answer

The first five terms are \( \frac{1}{2}, \frac{1}{6}, \frac{1}{24}, \frac{1}{120}, \frac{1}{720} \).

Step 1: Understand the sequence formula

The given sequence formula is \( a_n = \frac{1}{(n + 1)!} \). This means that each term of the sequence involves calculating the factorial of \( n + 1 \) and then taking the reciprocal of that factorial. The factorial of a number \( x \) is the product of all positive integers less than or equal to \( x \).

Step 2: Calculate the first term \(a_1\)

Set \( n = 1 \). Then, the first term is \( a_1 = \frac{1}{(1 + 1)!} = \frac{1}{2!} = \frac{1}{2} \).

Step 3: Calculate the second term \(a_2\)

Set \( n = 2 \). Then, the second term is \( a_2 = \frac{1}{(2 + 1)!} = \frac{1}{3!} = \frac{1}{6} \).

Step 4: Calculate the third term \(a_3\)

Set \( n = 3 \). Then, the third term is \( a_3 = \frac{1}{(3 + 1)!} = \frac{1}{4!} = \frac{1}{24} \).

Step 5: Calculate the fourth term \(a_4\)

Set \( n = 4 \). Then, the fourth term is \( a_4 = \frac{1}{(4 + 1)!} = \frac{1}{5!} = \frac{1}{120} \).

Step 6: Calculate the fifth term \(a_5\)

Set \( n = 5 \). Then, the fifth term is \( a_5 = \frac{1}{(5 + 1)!} = \frac{1}{6!} = \frac{1}{720} \).

Key Concepts

Factorial Function

The factorial function is a fundamental concept in mathematics, commonly denoted by an exclamation mark (!). It involves multiplying a series of descending natural numbers. For example, the factorial of 4, denoted as \( 4! \), is calculated as \( 4 \times 3 \times 2 \times 1 = 24 \).

Here are the key points about factorials:

• It is only defined for non-negative integers, starting from 0.
• The factorial of 0 is defined to be 1, i.e., \( 0! = 1 \).
• As numbers increase, their factorials grow very rapidly.
• This function is widely used in permutations, combinations, and other areas of mathematics.

Understanding factorials is essential when dealing with sequences, especially those involving factorial expressions as seen in our sequence formula.

Reciprocal

A reciprocal is another fundamental concept in mathematics. The reciprocal of a number \( x \) is simply \( \frac{1}{x} \). When multiplied with its reciprocal, the product is always 1.

Key aspects of reciprocals include:

• Differentiating between zero and non-zero numbers is crucial, as zero does not have a reciprocal.
• The reciprocal of a fraction \( \frac{a}{b} \) is \( \frac{b}{a} \).
• Reciprocals are frequently used in division, simplifying fractions, and solving equations.

In the context of our sequence, each term involves taking the reciprocal of a factorial, which provides a smaller fraction as the number of terms increases.

Sequence Formula

The sequence formula forms the backbone of a pattern or list of numbers. In our exercise, the formula is given by \( a_n = \frac{1}{(n + 1)!} \). This formula tells us precisely how to calculate each term in the sequence.

Important insights into sequence formulas:

• They define how each term in the sequence can be calculated.
• Typically, \( n \) represents the position of the term in the sequence.
• The sequence can be infinite or have a defined number of terms, depending on the context.
• Sequence formulas can involve various operations, such as addition, multiplication, or factorials, as seen here.

Understanding the sequence formula is crucial for determining any term within a sequence accurately.

Term Calculation

Term calculation involves using the sequence formula to find specific numbers in a sequence. This often requires substituting values into the formula, as shown in our exercise.

Steps in term calculation:

• Identify the position \( n \) of the term you want to calculate.
• Plug \( n \) into the sequence formula.
• Solve any operations, such as factorials or reciprocals, as needed.
• Simplify the result to find your term.

By following these steps, the first five terms of our sequence were accurately calculated as \( \frac{1}{2}, \frac{1}{6}, \frac{1}{24}, \frac{1}{120}, \) and \( \frac{1}{720} \), showcasing the power of a structured approach in mathematics.