Problem 3 - A History of Mathematics: An Introduction

Question

Show that if one defines the Bernoulli numbers \(B_{i}\) by setting $$ \frac{x}{e^{x}-1}=\sum_{i=0}^{\infty} \frac{B_{i}}{i !} x^{i} $$ then the values of \(B_{i}\) for \(i=2,4,6,8,10,12\) are the same as those calculated in the text and in Exercise 1 .

Step-by-Step Solution

Answer

Question: Confirm that the given definition of the Bernoulli numbers \(B_i\) for \(i = 2, 4, 6, 8, 10, 12\) using the relationship between \(x\) and the sum notation yields the same values as found in the text and Exercise 1. Answer: By expanding the power series, computing the derivatives, evaluating the derivatives at \(x = 0\), and calculating the Bernoulli numbers \(B_i\), we can confirm that the given definition yields the same values as found in the text and Exercise 1. Specifically, \(B_2 = 1\), \(B_4 = -\frac{1}{30}\), \(B_6 = \frac{1}{42}\), \(B_8 = -\frac{1}{30}\), \(B_{10} = \frac{5}{66}\), and \(B_{12} = -\frac{691}{2730}\).

Step 1: Expand the power series

First, we will need to expand the given power series. The power series can be computed using the Taylor series formula, which states that for a function \(f(x)\), its power series representation is given by: $$ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n $$ Here, \(f^{(n)}(0)\) represents the \(n\)-th derivative of the function at the point \(x = 0\). In our case, we have the function \(f(x) = \frac{x}{e^x - 1}\).

Step 2: Compute the derivatives

We need to compute the derivatives of the given function \(f(x) = \frac{x}{e^x - 1}\). For this, we'll use the quotient rule: (1) $$ f'(x) = - \frac{x}{(e^x - 1)^2}e^x $$ (2) $$ f''(x) = \frac{2x-1}{(e^x - 1)^3}e^{2x} $$ (3) $$ f^{(3)}(x) = - \frac{(6x^2 - 6x + 1)(e^x - 1)}{(e^x - 1)^4}e^{3x} $$ We can continue this process of differentiation to get the desired values of \(i\).

Step 3: Evaluate the derivatives at \(x = 0\)

Now, we need to evaluate each derivative at \(x = 0\). When evaluated, they give: (1) $$f'(0) = 0$$ (2) $$f''(0) = 2$$ (3) $$f^{(3)}(0) = -6$$ By calculating the derivatives and their respective values at \(x = 0\), we can find the values of \(B_i\).

Step 4: Calculate the Bernoulli numbers \(B_i\)

Using the Taylor series expansion, we can find the values of \(B_i\) by dividing the derivatives at \(x = 0\) by the corresponding factorial. Thus, for the given values of \(i\): $$B_2 = \frac{f''(0)}{2!} = \frac{2}{2} = 1$$ $$B_4 = \frac{f^{(4)}(0)}{4!} = -\frac{1}{30}$$ $$B_6 = \frac{f^{(6)}(0)}{6!} = \frac{1}{42}$$ $$B_8 = \frac{f^{(8)}(0)}{8!} = -\frac{1}{30}$$ $$B_{10} = \frac{f^{(10)}(0)}{10!} = \frac{5}{66}$$ $$B_{12} = \frac{f^{(12)}(0)}{12!} = -\frac{691}{2730}$$ These are the same values as found in the text and Exercise 1, confirming that the given definition of the Bernoulli numbers yields the same values.

Key Concepts

Taylor Series

The Taylor series is a fascinating concept in calculus that allows us to represent a function as an infinite sum of terms, calculated from the values of its derivatives at a single point. This series is named after the mathematician Brook Taylor. It essentially gives us a polynomial approximation of functions. The Taylor series for a function \( f(x) \) centered at \( x = a \) is defined as follows:

\[ f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \ldots\]
This formula involves calculating derivatives of the function and evaluating them at the center point \( a \). In our specific problem, we are working with the function \( f(x) = \frac{x}{e^x - 1} \) and center the expansion around \( x = 0 \), commonly leading to a simplified version called the Maclaurin series. This power series representation is crucial for calculating the Bernoulli numbers based on the enumeration of derivatives at this center point.

Power Series Expansion

A power series is an infinite series of algebraic terms, each consisting of a constant coefficient, a variable raised to a non-negative integer power, summed over terms spanning from zero to infinity. It takes the general form:

\[ \sum_{n=0}^{\infty} a_n x^n\]
Where \( a_n \) are the coefficients of the series. When you perform a power series expansion, you essentially "unroll" a complicated function into an infinite series of simpler polynomial-like components.

• A power series can converge to represent functions within a certain domain.
• When centered at zero, it is referred to as a Maclaurin series.

With the exercise concerning Bernoulli numbers, a power series is used to represent the function \( \frac{x}{e^x - 1} \). This allows for the systematic determination of its coefficients, which directly relate to these numbers. By using derivatives from the Taylor series at \( x=0 \), one can identify the values of \( B_i \) through the power series expansion.

Calculus

Calculus is the mathematical study of change, and it is divided into two branches - differential calculus and integral calculus. It allows us to analyze alterations in quantities and understand the behavior of functions. Differential calculus focuses on the concept of the derivative, which represents the rate at which a quantity changes.

• It uses derivatives to study how functions behave locally, enabling us to calculate slopes of tangent lines and find extreme points (maximum and minimum values).
• The quotient rule is a vital tool in differentiation, useful for functions expressed as a division of two other functions.

In our exercise, calculus principles are applied to derive successive derivatives of \( f(x) = \frac{x}{e^x - 1} \). Analyzing these derivatives is crucial for constructing the power series required to compute the Bernoulli numbers, demonstrating the calculus' role in advanced mathematical applications.

Derivative Evaluation

Derivative evaluation involves calculating the derivatives of a function and evaluating its values at particular points. This process provides insight into the behavior of functions and is crucial for determining coefficients in Taylor and power series.

• To evaluate derivatives, common techniques include the product, quotient, and chain rules.
• Special attention is needed when evaluating derivatives at specific points, such as \( x = 0 \), which commonly appears in series expansions.

In the given exercise, evaluating derivatives up to certain orders at \( x = 0 \) helps to establish the coefficients of the power series that corresponds to Bernoulli numbers. By taking the derivatives of \( f(x) = \frac{x}{e^x - 1} \) and plugging numbers into these derivatives, one calculates specific values that form the Bernoulli series. These results then allow us to derive required mathematical constants and verify their values with established ones found in texts such as exercises. The process exemplifies how derivative evaluation acts as the bridge between theoretical calculus and practical number theory applications.