Question

The factorial of a nonnegative integer \(n\) is written as \(n !\) (pronounced " \(n\) factorial") and is defined as follows: \(n !=n \cdot(n-1) \cdot(n-2) \cdot \ldots \cdot 1 \quad(\text { for values of } n \text { greater than or equal to } 1)\) 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 following formula. Allow the user to enter the number of terms to calculate. \(e=1+\frac{1}{1 !}+\frac{1}{2 !}+\frac{1}{3 !}+\ldots\) c) Write an application that computes the value of \(e^{x}\) by using the following formula. Allow the user to enter the number of terms to calculate. \(e^{x}=1+\frac{x}{1 !}+\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}+\ldots\)

Short answer

For (a), write a loop or recursive function to compute the factorial. For (b), develop code to estimate \(e\) using a finite series of \(1/n!\). For (c), code a program that calculates \(e^{x}\) using a series expansion with user-specified \(x\) and the number of series terms.

Step-by-step solution

Understanding Factorial

Understand that the factorial of a nonnegative integer \(n\), denoted as \(n!\), is the product of all positive integers from \(n\) down to 1. Specifically, \(n! = n \times (n-1) \times (n-2) \times \ldots \times 2 \times 1\). Special case: \(0! = 1\).

Writing a Factorial Application

Create an application (code or pseudo-code) with the capability to read a nonnegative integer \(n\), and then calculate its factorial using a loop or recursive function that multiplies the integers from \(n\) down to 1.

Understanding the Constant \(e\)

Recognize that the mathematical constant \(e\) can be estimated using the series \(e = 1 + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \ldots\). The more terms included, the better the approximation.

Writing an Application to Estimate \(e\)

Develop an application that calculates the value of \(e\) according to the series expansion given. It should allow the user to specify the number of terms to use in the calculation.

Extension to the Exponential Function

Understand that the power series for \(e^x\) is given by \(e^{x} = 1 + \frac{x}{1!} + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \ldots\).

Writing an Application to Compute \(e^{x}\)

Design an application that computes the value of \(e^{x}\) using the power series expansion. The application should allow the user to input the value of \(x\) and the number of terms for the series.

Key concepts

Java Programming

Java is an object-oriented programming language that's widely used for developing a variety of applications. To create a Java application capable of factorial computation, one starts by understanding the basics of Java syntax, including loops and methods.

For calculating a factorial in Java, you could use either an iterative approach with a 'for' loop or a recursive method that calls itself with the decremented value until reaching the base case, where the factorial of 0 is defined to be 1. It's critical to ensure handling of input to secure the application against entering negative numbers, as factorials for these are not defined.

In applications estimating the mathematical constant 'e' or computing exponential functions, similar looping structures are utilized where results of factorial computations are used in the series expansion for approximation. It's important to maintain precision in floating-point calculations and to understand how to control the loop iterations to achieve the desired accuracy.

Mathematical Constant e

The constant 'e' is a fundamental mathematical constant approximately equal to 2.71828, and is the base of the natural logarithm. This constant is irrational, which means it cannot be represented as a simple fraction and its decimal representation goes on forever without repeating.

The value of 'e' can be estimated using a factorial-based series expansion that adds the reciprocals of factorials starting from 0!. As more terms are summed, the estimate gets closer to the actual value of 'e'. This series expansion is a powerful example of how factorials are used in higher mathematics, not just in permutations and combinations but also in calculus and complex analysis.

The essence of creating an application to calculate 'e' is a lesson in both math and programming: it combines the concept of a limit (as the number of terms approaches infinity) with practical coding skills, such as loop control and output formatting in Java.

Exponential Function Computation

The exponential function, particularly when dealing with the constant 'e', is a crucial element in many areas of science and finance. Computing the value of e^x can be achieved by using a power series expansion where 'x' is raised to increasing powers and divided by the corresponding factorial.

In Java, this involves constructing a loop that runs for a number of terms specified by the user. In each iteration, the application calculates the term by raising 'x' to a power and then dividing by the factorial of that power. The sum of these terms approximates e^x. Care must be taken to manage precision and performance, as calculating factorials and powers can become computationally expensive.

Understanding the convergence properties of the series is important as well: with more terms included in the computation, the more precise the result will be. However, considering computing power and time, there's a practical limit to how many terms you should calculate.