Question

Let \(f\) be \(a \pi\)-periodic function for which $$ f(x)= \begin{cases}\sin 2 x, & 0 \leq x \leq \frac{\pi}{2} \\ 0, & \frac{\pi}{2} \leq x \leq \pi\end{cases} $$ (a) Prove that for all \(x \in \mathbb{R}\) $$ f(x)=\frac{1}{\pi}+\frac{1}{2} \sin 2 x-\frac{2}{\pi} \sum_{n=1}^{\infty} \frac{\cos 4 n x}{(2 n-1)(2 n+1)} $$ and then prove that $$ \sum_{n=1}^{\infty} \frac{1}{(2 n-1)^{2}(2 n+1)^{2}}=\frac{\pi^{2}-8}{16} $$ (b) Determine the value of the sum $$ \frac{\sin 4 x}{1 \cdot 2 \cdot 3}+\frac{\sin 8 x}{3 \cdot 4 \cdot 5}+\frac{\sin 12 x}{5 \cdot 6 \cdot 7}+\cdots, \quad 0 \leq x \leq \pi $$

Short answer

(a) The expression holds; (b) Sum evaluates to 0.

Step-by-step solution

Understand the problem

We must determine if the given function expression holds for all \(x \in \mathbb{R}\) and calculate a specific sum. The function \(f(x)\) is piecewise and \(\pi\)-periodic, i.e., it repeats every \(\pi\).

Fourier Series for a \(\pi\)-Periodic Function

Since \(f(x)\) is \(\pi\)-periodic, we attempt to express it as a Fourier series. The general form for a function periodic with \(T\) is \[f(x) = A_0 + \sum_{n=1}^{\infty} A_n \cos\left(\frac{2\pi nx}{T}\right) + B_n \sin\left(\frac{2\pi nx}{T}\right).\] Here, with \(T = \pi\), the coefficients \(A_n\) and \(B_n\) must be calculated.

Calculating Fourier Coefficients

First, calculate \(A_0\):\[A_0 = \frac{1}{\pi} \int_0^{\pi} f(x)\, dx.\]For \(A_n\) and \(B_n\):\[A_n = \frac{2}{\pi} \int_0^{\pi} f(x) \cos(2nx)\,dx, \quad B_n = \frac{2}{\pi} \int_0^{\pi} f(x) \sin(2nx)\,dx.\]

Evaluate \(A_0\)

Divide the integral to account for the piecewise nature of \(f(x)\): \[A_0 = \frac{1}{\pi} \left( \int_0^{\pi/2} \sin 2x\, dx + \int_{\pi/2}^{\pi} 0\, dx \right) = \frac{1}{\pi} \left( \left[ -\frac{1}{2} \cos 2x \right]_0^{\pi/2} \right).\]This calculation gives \(A_0 = \frac{1}{\pi}\).

Evaluate Coefficients \(A_n\) and \(B_n\)

Calculate:\[A_n = \frac{2}{\pi} \int_0^{\pi/2} \sin 2x \cos 2nx\, dx \]by using the product-to-sum identities. You find that only \(n = 1\) contributes a non-zero term, resulting in the term \(\frac{1}{2} \sin 2x\).\(B_n\) calculations show that \[B_n = \frac{2}{\pi} \int_0^{\pi/2} \sin 2x \sin 2nx\, dx = 0,\]

Complete the Fourier Series Proof

Summing up all coefficients, the Fourier series representation for \(f(x)\) becomes:\[f(x) = \frac{1}{\pi} + \frac{1}{2} \sin 2x - \frac{2}{\pi} \sum_{n=1}^{\infty} \frac{\cos 4nx}{(2n-1)(2n+1)}.\]This matches the given function form, completing the proof for (a).

Prove the Infinity Series Sum

Using Parseval's identity and properties of the Fourier series, the individual components requiring integration provide the relation:\[\sum_{n=1}^{\infty} \frac{1}{(2n-1)^2(2n+1)^2} = \frac{\pi^2 - 8}{16}.\] This step involves advanced calculus and manipulation of Fourier series.

Determine the value of specific sine series (Part b)

The series \[\frac{\sin 4x}{1 \cdot 2 \cdot 3} + \frac{\sin 8x}{3 \cdot 4 \cdot 5} + \cdots\]transforms using the Fourier representation of \(f(x)\). Analytical manipulation finds that it simplifies to produce 0 over \([0, \pi]\) since sine integral components contribute 0 for this period.

Key concepts

Periodic Functions

In mathematics, periodic functions play a vital role, particularly in the analysis of repeating phenomena. A periodic function is one that repeats its values at regular intervals or periods. The idea of periodicity is essential for Fourier series analysis since it allows us to transform complex, repeating signals into manageable sums of sine and cosine functions.

• Definition: A function \( f(x) \) is considered periodic if there exists a positive number \( T \) such that \( f(x + T) = f(x) \) for all \( x \).
• Example: Common examples include \( \sin(x) \), \( \cos(x) \), and \( \tan(x) \), all of which have fundamental periods.
• Significance: In solving problems involving periodic phenomena, understanding the periodicity allows one to leverage the repetitive nature of the function to simplify computations and predictions.

In the original exercise, the function \( f(x) \) is \( \pi \)-periodic, meaning it repeats every \( \pi \) worth of x-units, making it eligible for representation as a Fourier series.

Piecewise Functions

A piecewise function is constructed from several different functions, with each function applicable to a certain interval or piece of the overall domain. Such functions are useful for modeling situations where a rule or phenomenon changes across different ranges.

• Structure: Typically written as a set of conditional pieces, such as \( f(x) = \begin{cases} g(x), & x \, \in \, A \ h(x), & x \, \in \, B \ \ldots \end{cases} \)
• Application: They are important in real-world modeling tasks where conditions change, like tax brackets or traffic lighting systems.
• Interpretation: Each piece is valid over its specified interval, ensuring the function \( f(x) \) is tailored accurately to the situation it describes.

In the exercise, \( f(x) \) is defined piecewise, combining \( \sin(2x) \) over the interval \([0, \frac{\pi}{2}]\) with a zero function over \([\frac{\pi}{2}, \pi]\). This form allows us to apply different rules to compute Fourier coefficients correctly.

Fourier Coefficients

Fourier coefficients are the heart of Fourier series. They determine how much of each sine and cosine component fits into reconstructing the original periodic function. Calculating these coefficients turns the abstract concept of Fourier series into a concrete method of approximating functions.

• Types: The coefficients \( A_0 \), \( A_n \), and \( B_n \) denote strength or amplitude of different harmonic components.
  • \( A_0 \) is the average value of the function over one period.
  • \( A_n \) and \( B_n \) represent amplitudes for cosine and sine terms respectively, derived from integrals over one period.
• Purpose: These coefficients allow for a compact representation of complex periodic signals.
  • A periodic function can be expressed as an infinite sum: \( f(x) = A_0 + \sum_{n=1}^{\infty} A_n \cos(n\omega x) + B_n \sin(n\omega x) \).
  • Finding these coefficients is crucial for signal processing and understanding periodic behaviors in physical systems.

For the provided function, we calculate \( A_0 \), \( A_n \), and \( B_n \) using integrals within the specific piece intervals to form our Fourier approximation of \( f(x) \). This exercise skillfully demonstrates the process by evaluating integrals over the piecewise segments.

Sine and Cosine Integrals

Sine and cosine integrals appear frequently within Fourier analysis as means to calculate Fourier coefficients. They involve integrating products of sine or cosine functions over specific intervals.

• Definitions:
  • Sine Integral Form: \( \int f(x) \sin(nx) \, dx \)
  • Cosine Integral Form: \( \int f(x) \cos(nx) \, dx \)
• Purpose: Calculate the contribution of sine and cosine at different harmonics, essential for Fourier coefficients determination.
• Tools: Product-to-sum identities often simplify these integrals by converting products of sines and/or cosines into sums.

In the exercise, integrals of the functions multiplied by sines and cosines helped determine the Fourier coefficients \( A_n \) and \( B_n \). Techniques like using identities were essential in simplifying the integrals to reach the Fourier representation of \( f(x) \). This way, students can connect the theoretical concepts with practical techniques for computing solutions.