Problem 2 - Thomas Calculus

Question

Each of Exercises \(1-6\) gives a formula for the \(n\) th term \(a_{n}\) of a sequence \(\left\\{a_{n}\right\\} .\) Find the values of \(a_{1}, a_{2}, a_{3},\) and \(a_{4} .\) $$ a_{n}=\frac{1}{n !} $$

Step-by-Step Solution

Answer

\(a_1 = 1\), \(a_2 = \frac{1}{2}\), \(a_3 = \frac{1}{6}\), \(a_4 = \frac{1}{24}\).

Step 1: Identify the Formula

The formula given for the sequence is \( a_{n} = \frac{1}{n!} \). This indicates that each term in the sequence is the reciprocal of the factorial of \( n \).

Step 2: Calculate \( a_1 \)

To find \( a_1 \), substitute \( n = 1 \) into the formula: \[ a_1 = \frac{1}{1!} = \frac{1}{1} = 1 \]

Step 3: Calculate \( a_2 \)

To find \( a_2 \), substitute \( n = 2 \) into the formula: \[ a_2 = \frac{1}{2!} = \frac{1}{2 \times 1} = \frac{1}{2} \]

Step 4: Calculate \( a_3 \)

To find \( a_3 \), substitute \( n = 3 \) into the formula: \[ a_3 = \frac{1}{3!} = \frac{1}{3 \times 2 \times 1} = \frac{1}{6} \]

Step 5: Calculate \( a_4 \)

To find \( a_4 \), substitute \( n = 4 \) into the formula: \[ a_4 = \frac{1}{4!} = \frac{1}{4 \times 3 \times 2 \times 1} = \frac{1}{24} \]

Key Concepts

Factorial

The concept of a factorial is fundamental in mathematics, especially when dealing with sequences and permutations. A factorial, denoted by the exclamation mark "!", is the product of all positive integers less than or equal to a number.

• For example, the factorial of 4, written as \(4!\), is calculated as \(4 \times 3 \times 2 \times 1 = 24\).
• The factorial of 0 is defined as 1, i.e., \(0! = 1\).
• Factorials grow very quickly with increasing values of \(n\).

This concept appears frequently in sequences like our original exercise, where the \(n\)th term is \(\frac{1}{n!}\). Calculating with factorials often involves recognizing patterns in multiplying consecutive numbers. Factorials facilitate understanding complex mathematical relationships, making them essential in areas like calculus and algebra.

Recurrence relation

A recurrence relation is an equation or inequality that defines a sequence based on the previous terms.

• It is particularly useful in computer science and mathematics to model sequences or series.
• Recurrence relations can define simple sequences, such as arithmetic or geometric progressions, and more complex structures like the Fibonacci sequence.

Unlike the formula \(a_n = \frac{1}{n!}\), a recurrence relation would express \(a_n\) in terms of \(a_{n-1}\), \(a_{n-2}\), etc. For example, in a simple recurrence, you might have \(a_n = a_{n-1} + a_{n-2}\) for certain sequences. Recurrence relations are vital for iterative algorithms, as they provide a clear pathway from one state to the next. By understanding how each term is linked to others, we can predict the behavior of a sequence or series.

Series

A series is a sum of the terms of a sequence. When you add each term of a sequence, the resulting sum is called a series.

• One common example is an arithmetic series, where the difference between consecutive terms is consistent.
• Another is a geometric series, where each term is a fixed multiple of the previous one.
• Series can be finite or infinite.

Consider the sequence given by \(a_n = \frac{1}{n!}\). If we add these terms, such as \(1 + \frac{1}{2} + \frac{1}{6} + \frac{1}{24} + \ldots\), we create a series. Each additional term brings the partial sum closer to a particular value. Understanding series is critical in calculus and real-world applications like finance, where future values are predicted using the summation of terms over time. The convergence or divergence of a series can tell much about its overall behavior.