Question

We can get an approximate value for \(e\) from the following infinite series:$$\begin{aligned} &\begin{array}{l}\text { Series } \\\\\text { Approximation } \quad e=2+\frac{1}{2 !}+\frac{1}{3 !}+\frac{1}{4 !}+\ldots \quad 158 \\\\\text { for }e\end{array}\\\&\end{aligned}$$.where \(4 !\) (read " 4 factorial") is \(4(3)(2)(1)=24\). Write a program to compute \(e\) using the first five terms of the series.

Short answer

The approximate value of e using the first five terms of the series is 3.7084.

Step-by-step solution

Understanding the Series

The given series for the approximate value of e is an infinite series where each term after the first is the reciprocal of the factorial of n, for n starting at 1. The term 'n!' denotes the factorial of n, which is the product of all positive integers up to n. To approximate e, we need to sum the first five terms of the series which are: 1, 1/1!, 1/2!, 1/3!, and 1/4!.

Calculating the Factorials

Before summing the series, we need to calculate the factorials for the terms. Here are the calculations: 1! = 1, 2! = 2(1) = 2, 3! = 3(2)(1) = 6, and 4! = 4(3)(2)(1) = 24. We already know that 1! equals 1 by definition.

Summing the First Five Terms

Next, we calculate the first five terms of the series: Term 1 is 1 (which is 2, as given), Term 2 is 1/1! = 1, Term 3 is 1/2! = 1/2, Term 4 is 1/3! = 1/6, and Term 5 is 1/4! = 1/24. To find the approximate value of e, we sum these terms.

Computing the Approximation

The approximate value of e using the first five terms is therefore: e ≈ 2 + 1/1! + 1/2! + 1/3! + 1/4! = 2 + 1 + 1/2 + 1/6 + 1/24. Simplify the expression by finding a common denominator or by direct addition of the fractions.

Simplified Approximation

The simplified form of the approximation, by adding the fractions directly, is: e ≈ 2 + 1 + 0.5 + 0.1667 + 0.0417 = 3.7084. Thus, the value of e using the first five terms of the series is approximately 3.7084.

Key concepts

Factorial

When we speak about factorials in mathematics, we refer to a special operation that multiplies a series of descending positive integers. The factorial of a number 'n', denoted as 'n!', is the product of all positive integers from 'n' down to 1. For instance,\( 4! = 4 \times 3 \times 2 \times 1 = 24 \).

Factorials grow very rapidly with each additional number, which makes them significant in various areas of mathematics, including combinatorics, algebra, and calculus, as they are used to calculate permutations and combinations, series, and more. It's important to note that the factorial of zero is defined as 1, or \( 0! = 1 \). Not only does this definition make mathematical sense, but it also helps maintain consistency in equations and formulas involving factorials.

Infinite Series

An infinite series is a summation of a sequence of numbers that, as the name suggests, continues indefinitely. In mathematics, an infinite series is often written as \( a_1 + a_2 + a_3 + ... \), where the \( a_i \) are the terms in the series. A notable characteristic of infinite series is that they can converge or diverge. Convergence means that as more and more terms are added, the sum approaches a finite limit. On the other hand, divergence means the series does not approach a particular value as more terms are added.

The series depicting Euler's number 'e' is an example of an infinite series that converges. While adding more terms gives a more accurate approximation, even a finite number of terms can provide a useful estimate. In this case, using the first few terms provides a reasonable estimate of 'e', which is especially practical in computations where exact values are not required or are difficult to obtain.

Mathematical Approximation

Approximation is a fundamental concept in mathematics, essential when an exact answer is either impossible or unnecessary. It involves finding a value that is close enough to the correct answer for the purpose at hand. Approximations can come in many forms, such as rounding decimals, estimating pi, or truncating infinite series.

In the context of the exercise, we see approximation at work with the infinite series for Euler's number 'e'. Using only the first five terms gives us an approximate value which becomes increasingly accurate with the addition of more terms. Mathematical approximations such as this are invaluable in scientific computations, engineering, and statistical analysis, among other fields. They enable us to work with complex numbers and concepts in a manageable and pragmatic way.

Euler's Number

Euler's number, denoted 'e', is one of the most important constants in mathematics, much like pi (\( \pi \)). This irrational number is approximately equal to 2.71828 and is the base of the natural logarithm. The numerical value of 'e' arises in many contexts, such as growth processes, compound interest calculations, and in describing certain probability distributions.

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