Question

The following simple computation yields a good approximation to Stirling's formula. For \(m=1,2,3, \ldots\), define $$ f(x)=(m+1-x) \log m+(x-m) \log (m+1) $$ if \(m \leq x \leq m+1\), and define $$ g(x)=\frac{x}{m}-1+\log m $$ if \(m-\frac{1}{2} \leq x-t+\int_{1}^{x} g(x) d x . $$ Integrate \(\log x\) over \([1, n]\). Conclude that $$ z<\log (n l)-(n+f) \log n+n<1 $$ for \(n=2,3,4, \ldots\) (Note: \(\log \sqrt{2 \pi} \sim 0.918 \ldots .)\) Thus $$ e^{7 / t}<\frac{n !}{(n / e)^{4} \sqrt{n}}

Short answer

Answer: The inequality concluded for approximating factorials using Stirling's formula is: $$ e^{7 / t}<\frac{n !}{(n / e)^{4} \sqrt{n}}<e $$

Step-by-step solution

Understand the functions

We are given two functions, \(f(x)\) and \(g(x)\), that will help us approximate Stirling's formula. \(f(x)\) is defined for \(m \leq x \leq m + 1\) and \(g(x)\) is defined for \(m - \frac{1}{2} \leq x < m + 1\). We also have that \(f(x) \leq \log x \leq g(x)\) for \(x \geq 1\).

Draw the graphs of f(x) and g(x)

Draw the graphs of \(f(x)\) and \(g(x)\) on the same plane. Notice the shape of the graphs and how they are related to each other, with \(f(x)\) being under the graph of \(g(x)\) for the given domain.

Integrate f(x) and g(x)

Now, we are given the following integral inequalities: $$ \int_{1}^{n} f(x) d x=\log (n !)-\downarrow \log n>-t+\int_{1}^{x} g(x) d x . $$ Find the integrals for both \(f(x)\) and \(g(x)\) over the given domain.

Integrate log(x) over [1, n]

Integrate \(\log x\) over the interval \([1, n]\). Using the inequalities found in Step 3, we now have an equation that involves the integral of \(\log x\).

Conclude the inequality for n!

From the previous steps, we have enough information to conclude the inequality: $$ z<\log (n l)-(n+f) \log n+n<1 $$ for \(n=2,3,4, \ldots\).

Relate inequality to Stirling's formula

The inequality found states that: $$ e^{7 / t}<\frac{n !}{(n / e)^{4} \sqrt{n}}<e $$ This is our final result, and it gives us a way to approximate factorials using Stirling's formula. Note that \(\log \sqrt{2 \pi} \approx 0.918\), which further solidifies the approximation.

Key concepts

Mathematical Analysis

Mathematical analysis is a branch of mathematics that deals with understanding the changes in quantities and their relationships. It provides a rigorous foundation for the concepts of limits, continuity, and infinite series. In our context, mathematical analysis comes into play when we're working with functions and integrals to approximate factorials using Stirling's formula.

In the exercise provided, the functions f(x) and g(x) are constructed to provide bounds for the logarithm of x, and the integral of these functions over a range creates a bridge to approximating a factorial. Understanding the behavior of these functions and their relationship to \(\log x\) is crucial in analyzing how good the approximation is. Through mathematical analysis, we can make sense of the integral inequalities presented, which are pivotal in leading us to an approximation of Stirling's formula.

Integral Inequalities

Integral inequalities are an important part of calculus, which establish the relationship between functions over an interval. Specifically, they can show us how one function can serve as a lower or upper bound for another when integrated over the same interval. This concept is central to the exercise involving Stirling's formula.

The step-by-step solution shows us that f(x) lies below \(\log x\) and \(\log x\) lies below g(x) within the specified domain. Through the integration of these functions, we are allowed to create bounds for the integral of \(\log x\), which is connected to the logarithm of n! Through such inequalities, we can make rigorous statements about the relationship between these integrals and use them to estimate with a degree of certainty.

Factorial Approximation

Factorials grow at a rapid rate, making them difficult to compute or estimate for large values of n. Stirling's formula is a means of approximating these large factorials, instrumental in the fields of combinatorics, probability, and statistical mechanics, among others. It is crucial for students to understand the mechanics behind Stirling's formula as it simplifies complex calculations.

In the exercise, the factorial approximation is derived from integrating logarithmic functions and applying integral inequalities. The idea is that the logarithm of a factorial can be bounded by simpler expressions that, when exponentiated, yield an approximation of the factorial itself. The relationship demonstrated between \(n!\), \((n/e)^n\), and \(n^{1/2}\) illustrates the application of Stirling's formula and reaffirms its importance in mathematical computation.

Logarithmic Integration

Logarithmic integration involves finding the integral of the logarithmic function, which often arises in problems related to growth and decay. Logarithms transform multiplicative relationships into additive ones, which can simplify the process of integration. In the context of Stirling's formula, the integral of the \(\log x\) function links to the behavior of factorials.

Demonstrating the integration of \(\log x\) plays a pivotal role in the exercise, allowing us to estimate \(\log(n!)\) via integral inequalities. By integrating \(\log x\) from 1 to n, we can estimate the sum of logs of all integers up to n, which is equivalent to the \(\log(n!)\). This technique is a powerful tool that bridges the discrete nature of factorial growth with the continuous methods of calculus, and it offers a deeper understanding of the relationship between the two through mathematical analysis.