Question

Do the following: a. Find the first four nonvanishing terms of the Maclaurin series expansion of \(f(x)=\frac{x}{e^{x}-1} .\) b. Use the result in part a. to determine the first four nonvanishing Bernoulli numbers, \(B_{n}\). c. Use these results to compute \(\zeta(2 n)\) for \(n=1,2,3,4\).

Short answer

The first four non-vanishing terms of the Maclaurin series expansion of \(f(x)=\frac{x}{e^{x}-1}\) are \(x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \frac{x^{7}}{7!}\). The first four non-vanishing Bernoulli numbers, \(B_{n}\), are \(B_{0} = 1\), \(B_{1} = -1/2\), \(B_{2} = 1/6\), \(B_{4} = -1/30\). The values of \(\zeta(2 n)\) for \(n=1,2,3,4\) are given by \(\zeta(2) = \pi^2/6\), \(\zeta(4) = \pi^4/90\), \(\zeta(6) = \pi^6/945\), \(\zeta(8) = \pi^8/9450\).

Step-by-step solution

Finding the Maclaurin Series Expansion

Start with the given function \(f(x) = \frac{x}{e^{x}-1}\). We will determine its expansion by calculating the derivatives and putting them into Taylor's formula. The Maclaurin series is a specific kind of Taylor series where we are applying the formula around the point \(x=0\). With the regularity conditions and derivatives, the first four non-vanishing terms in the Maclaurin series can be calculated as \(x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \frac{x^{7}}{7!}\).

Calculation of Bernoulli Numbers

The Bernoulli numbers can be calculated using the Maclaurin series expansion. The nth Bernoulli number is given by the coefficient of \(x^{n}\) in the expansion of \(x / (e^{x} - 1)\). Comparing the terms in the expansion, we find: \(B_{0} = 1\), \(B_{1} = -1/2\), \(B_{2} = 1/6\), and \(B_{3} = 0\). Hence, the first four non-vanishing Bernoulli numbers are \(B_{0} = 1\), \(B_{1} = -1/2\), \(B_{2} = 1/6\), and \(B_{4} = -1/30\) (since \(B_{3} = 0\) is a vanishing term).

Calculation of Zeta Function

The Riemann Zeta function can be computed using the Bernoulli numbers. The function \(\zeta(2 n)\) is given by \(\zeta(2 n) = (-1)^(n+1) (2\pi)^{2n} B_{2n}/(2(2n)!)\). Substituting the values for \(n=1,2,3,4\) gives us the following: \(\zeta(2) = \pi^2/6\), \(\zeta(4) = \pi^4/90\), \(\zeta(6) = \pi^6/945\), \(\zeta(8) = \pi^8/9450\).

Key concepts

Riemann Zeta Function

The Riemann Zeta function, denoted as \( \zeta(s) \), is a complex function that appears in many areas of mathematics, most notably in number theory and complex analysis. It is defined for complex numbers \(seq1\) with a real part greater than 1 by the infinite series \[ \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} \]. However, through analytic continuation, it can be extended to other values of \(s\), except for \(s=1\), where it has a simple pole.

For positive even integers, the Riemann Zeta function has a surprising connection to \(\pi\) (Pi), as seen in the formulas for \(\zeta(2n)\) using the Bernoulli numbers, which makes it a powerful tool in number theory. It's also intimately related to the distribution of prime numbers via the Riemann Hypothesis, one of the most famous unsolved problems in mathematics.

When used in exercises like the one provided, the Riemann Zeta function becomes more concrete as students learn to compute specific values for \(\zeta(2n)\) and see the interplay between analysis and number theory.

Bernoulli Numbers

Bernoulli numbers are a sequence of rational numbers that have deep implications in various branches of mathematics, including number theory, analysis, and the topology of manifolds. They appear in the Taylor series expansions of many trigonometric and exponential functions and are critical for calculating the Riemann Zeta function values at positive even integers.

By playing around with the Maclaurin series, students can see firsthand how Bernoulli numbers are used in practice. They emerge as the coefficients in the series expansion of the function \( \frac{x}{e^x - 1} \) around \(x = 0\). For instance, if we follow the problem-solving steps for finding the first four nonvanishing Bernoulli numbers, we unveil a pattern that can be generalized to find higher-order Bernoulli numbers, enhancing students' understanding of the concept.

It is noteworthy that Bernoulli numbers are not just theoretical constructs; they are practical tools in computing sums of powers and in the study of numerical series, making them an essential concept for students to grasp.

Taylor Series

The Taylor series represents a function as an infinite sum of terms calculated from the values of its derivatives at a single point. It's a powerful tool for approximating complex functions with a series of polynomial terms, which are much simpler to work with. The Maclaurin series is a special case of the Taylor series, where the expansion takes place around zero \(x = 0\).

Understanding the Taylor series, as well as its importance in mathematics, is crucial for students tackling calculus and analysis. The series not only provides a way to approximate functions but also helps to understand the behavior and properties of functions near a point. In the step-by-step solution for the problem, the Maclaurin series is used to break down a complex function into more manageable parts, demonstrating an essential application of the Taylor series in solving real-world problems.

By incorporating exercise improvement advice, students are encouraged to delve deeper into the inner workings of the Taylor series, such as manipulating and differentiating functions to find the series expansion, offering a practical approach to learning this key mathematical concept.