Question

The factorial of a nonnegative integer \(n\) is written as \(n !\) (pronounced "n factorial") and is defined as follows: \(n !=n \cdot(n-1) \cdot(n-2) \dots \dots 1 \quad \text { (for values of } n \text { greater than or equal to } 1)\) and \\[ n !=1 \quad(\text { for } n=0) \\] For example, \(5 !=5 \cdot 4 \cdot 3 \cdot 2 \cdot 1,\) which is 120 a) Write an application that reads a nonnegative integer and computes and prints its factorial. b) Write an application that estimates the value of the mathematical constant \(e\) by using the formula \\[ e=1+\frac{1}{1 !}+\frac{1}{2 !}+\frac{1}{3 !}+\dots \\] c) Write an application that compures the value of \(e^{x}\) by using the formula \\[ e^{x}=1+\frac{x}{1 !}+\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}+\ldots \\]

Short answer

Use loops to compute factorials and series to estimate \(e\) and \(e^x\).

Step-by-step solution

Understanding the Factorial Function

The factorial of a non-negative integer \( n \) is the product of all positive integers less than or equal to \( n \). This is expressed as \( n! \), which is defined as follows: \( n! = n \cdot (n-1) \cdot (n-2) \dots 1 \). For instance, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). Additionally, by definition, \( 0! = 1 \). This is the foundational concept for solving the problems.

Calculating Factorials

For part (a), write a program that prompts the user to enter a nonnegative integer and then calculates its factorial. You can use a loop to multiply the numbers decrementally until 1. For example, starting from \( n \), multiply \( (n-1), (n-2), ..., \) until you reach 1.

Estimating the Value of e

For part (b), create a program that uses the sum of the series \( 1 + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \, \ldots \) to estimate the mathematical constant \( e \). You can compute this by iterating through terms of this series and adding them together until a predefined level of accuracy is reached, which will depend on the number of terms you decide to sum.

Calculating e^x Using Series

For part (c), write a program to compute \( e^x \) using the series \( 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \, \ldots \). This approach is similar to the estimation of \( e \), but instead of just \( 1/n! \), each term is \( x^k/k! \) where \( k \) is the term index starting from 0. The series allows calculating \( e^x \) for different values of \( x \) by increasing number of terms until accuracy is achieved.

Key concepts

Mathematical Constant e

The mathematical constant \( e \) is a fundamental number in mathematics, similar in importance to \( \pi \). This constant, approximately equal to 2.71828, is the base of natural logarithms and is a key component in calculus, complex numbers, and exponential growth problems.

One fascinating property of \( e \) is that it can be defined using an infinite series. We can approximate \( e \) by adding up a series of terms, as shown by the formula:

• \( e = 1 + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \ldots \)

This series continues indefinitely, and with each term, the approximation becomes closer to the actual value of \( e \).

The understanding of \( e \) is essential as it helps us describe patterns of growth, primes us for deeper insights in calculus, and provides a standard framework for exponential functions.

Series Calculation

Series calculation is a way of adding a sequence of numbers, called terms, to find their total. In estimating the value of constants like \( e \) or computing functions like \( e^x \), series calculations become very powerful.

A series typically is expressed as a list of terms separated by plus signs. The infinite series for \( e \) converges rapidly, meaning that even a few terms can give a good approximation. When computing \( e \) from its series, each added term \( \frac{1}{n!} \) contributes less significantly to the total sum, making it a very efficient way to approximate \( e \).

In practice, you decide how many terms to use for a series, balancing between computational effort and the desired precision. This decision depends on context, such as the level of accuracy needed for scientific calculations or real-world applications.

Exponential Function e^x

The exponential function, denoted as \( e^x \), extends the concept of the constant \( e \) into exponential growth and decay processes. This function is significant in many fields, including finance, physics, and population dynamics.

To calculate \( e^x \), we utilize a power series, which is a sum of terms that take the form of charged powers of \( x \) divided by factorials. The series is represented like this:

• \( e^x = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \ldots \)

This series similar to that of \( e \), but each term is affected by \( x \), making it flexible for different values.

Understanding this function allows us to model natural processes mathematically. Since \( e^x \) is used to describe growth and decay, it is a staple in applications that involve compound interest, population studies, or radioactive decay, to name a few. By calculating \( e^x \) using its series, we get an efficient and precise way to understand exponential relationships.