Question

Show that the Fourier series for \(|\sin \theta|\) in the range \(-\pi \leq \theta \leq \pi\) is given by $$ |\sin \theta|=\frac{2}{\pi}-\frac{4}{\pi} \sum_{m=1}^{\infty} \frac{\cos 2 m \theta}{4 m^{2}-1} $$ By setting \(\theta=0\) and \(\theta=\pi / 2\), deduce values for $$ \sum_{m=1}^{\infty} \frac{1}{4 m^{2}-1} \quad \text { and } \sum_{m=1}^{\infty} \frac{1}{16 m^{2}-1} $$

Short answer

\( \sum_{m=1}^{\infty} \frac{1}{4 m^{2}-1} = \frac{1}{2}\) and \ \sum_{m=1}^{\infty} \frac{1}{16 m^{2}-1} = -\frac{1}{4} \

Step-by-step solution

Define the Fourier series for \(|\sin \theta|\)

The Fourier series for a function can be expressed as the sum of sines and cosines (or complex exponentials). For \(|\sin \theta|\) in the interval \(-\pi \leq \theta \leq \pi\), the series expansion is given: \[ |\sin \theta| = \frac{2}{\pi} - \frac{4}{\pi} \sum_{m=1}^{\infty} \frac{\cos 2m \theta}{4 m^{2}-1} \]

Set \(\theta = 0\)

Substitute \(\theta = 0\) into the Fourier series for \(|\sin \theta|\) and simplify: \[ |\sin 0| = \frac{2}{\pi} - \frac{4}{\pi} \sum_{m=1}^{\infty} \frac{\cos(0)}{4 m^{2}-1}\]We know that |\sin 0| = 0 and \( \cos(0) = 1 \), so:\[ 0 = \frac{2}{\pi} - \frac{4}{\pi} \sum_{m=1}^{\infty} \frac{1}{4 m^{2}-1}\]Thus, solving for the series:\[ \sum_{m=1}^{\infty} \frac{1}{4 m^{2}-1} = \frac{1}{2} \]

Set \(\theta = \pi / 2\)

Substitute \(\theta = \pi / 2\) into the Fourier series for \(|\sin \theta|\) and simplify: \[ |\sin \frac{\pi}{2}| = \frac{2}{\pi} - \frac{4}{\pi} \sum_{m=1}^{\infty} \frac{\cos( \pi m)}{4 m^{2}-1} \]We know that |\sin(\pi / 2)| = 1 and \( \cos(\pi m) = (-1)^m \), so: \[ 1 = \frac{2}{\pi} - \frac{4}{\pi} \sum_{m=1}^{\infty} \frac{(-1)^m}{4 m^{2}-1} \]Thus solving for the series:\[ \sum_{m=1}^{\infty} \frac{(-1)^m}{4 m^{2}-1} = -\frac{1}{4} \]

Key concepts

Fourier analysis

Fourier analysis is a powerful mathematical tool used to break down complex periodic signals into simpler components. It decomposes a function into a sum of sines and cosines, which are easier to analyze.
This is particularly useful in many fields such as signal processing, physics, and engineering.
In this specific example, we are tasked with finding the Fourier series for \(|\sin \theta|\) within the interval \-\(\text{{pi}} \leq \theta \leq \text{{pi}}\). This involves expressing \(|\sin \theta|\) as a series of cosine terms. Techniques like this help in understanding the frequency components of a given function.

Trigonometric series

A trigonometric series is a series of functions that involves sines and cosines. These series are crucial in representing periodic functions.
The Fourier series is a type of trigonometric series.

• In our exercise, we express \(|\sin \theta|\) as a Fourier series consisting of cosine terms.
• The general form looks like: \[|\text{{sin}} \theta| = \frac{2}{\text{{pi}}} - \frac{4}{\text{{pi}}} \sum_{{m=1}}^{{\infty}} \frac{\cos(2m\theta)}{{4m^2-1}} \] Understanding such series helps in simplifying the study of periodic functions.
  Whether you're dealing with tides, sound waves, or electrical signals, trigonometric series are an invaluable tool.

Convergence of series

The concept of convergence is critical for working with series.
A series converges if the sum of its terms approaches a specific limit as more terms are added.
In our Fourier series for \(|\sin \theta|\), convergence is essential to ensure that the series accurately represents the function. Without convergence, the series would not provide a reliable approximation.

Mathematical proofs

Mathematical proofs provide a logical foundation for establishing the truth of a statement.
Proving the series for \(|\sin \theta|\) involves careful substitution and simplification to ensure correctness. In our exercise, we perform the steps:

• Substitute \(\theta = 0\) into the Fourier series formula and simplify.
• Substitute \(\theta = \pi / 2\) and simplify again.

Such proofs validate the general formula and ensure our results are accurate.
They also help in finding the sums of specific series as demonstrated in our solutions.

Sum of series

Computing the sum of a series is an important aspect of analysis.
In this exercise, we deduce the values for:

• \[\sum_{{m=1}}^{{\infty}} \frac{1}{{4m^2-1}} = \frac{1}{2} \]
• \[\sum_{{m=1}}^{{\infty}} \frac{1}{{16m^2-1}} = \frac{1}{4} \]

These sums are obtained by substituting specific values of \(\theta\) into the Fourier series and solving.
This highlights the practical application and power of Fourier series in solving mathematical problems related to sums and series.