Question

Set \(f(x)=|\sin x|\) and let $$ f(x) \sim \frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left[a_{n} \cos n x+b_{n} \sin n x\right] $$ denote the Fourier series of \(f\) on \([-\pi, \pi]\). (a) Calculate the \(a_{n}\) and \(b_{n}\). (b) Set \(g(x)=\sum_{n=1}^{\infty}\left[-n a_{n} \sin n x+n b_{n} \cos n x\right]\). Determine \(g\) and sketch the graph of \(g\) on \([-\pi, \pi]\). (c) Calculate the sums $$ \sum_{n=1}^{\infty} \frac{1}{4 n^{2}-1}, \quad \sum_{n=1}^{\infty} \frac{(-1)^{n}}{4 n^{2}-1}, \quad \sum_{n=1}^{\infty} \frac{1}{\left(4 n^{2}-1\right)^{2}}, \quad \sum_{n=1}^{\infty} \frac{n^{2}}{\left(4 n^{2}-1\right)^{2}} . $$

Short answer

(a) \(a_n = \frac{4(-1)^{n+1}}{\pi(1-n^2)}\), \(b_n = 0\). (b) \(g(x)\) is an oscillating function. (c) \(\frac{1}{2}, \frac{1}{4}, \frac{3}{16}, \frac{1}{8}\).

Step-by-step solution

Find Fourier Coefficients

The Fourier coefficients for the function \( f(x) = |\sin x| \) are given by:1. For \( a_0 \): \[a_0 = \frac{1}{\pi}\int_{-\pi}^{\pi} |\sin x| \, dx = 2 \cdot \frac{1}{\pi} \int_{0}^{\pi} \sin x \, dx = \frac{4}{\pi}\] 2. For \( a_n \) (n \geq 1):\[a_n = \frac{1}{\pi}\int_{-\pi}^{\pi} |\sin x| \cos(nx) \, dx = \frac{2}{\pi}\int_{0}^{\pi} \sin x \cos(nx) \, dx \]Using integration by parts, the evaluation gives:\[a_n = \frac{4(-1)^{n+1}}{\pi(1-n^2)}\]3. For \( b_n \): \[b_n = \frac{1}{\pi}\int_{-\pi}^{\pi} |\sin x| \sin(n x) \, dx = 0\]This is because \(|\sin x|\) is an even function and \(\sin(n x)\) is an odd function.

Express g(x) and Simplify

Given \(g(x) = \sum_{n=1}^{\infty} [-n a_n \sin nx + n b_n \cos nx]\) and since \(b_n = 0\) for all \(n\), the function simplifies to:\[g(x) = -\sum_{n=1}^{\infty} n a_n \sin nx\]Substituting \(a_n\), we get:\[g(x) = -\sum_{n=1}^{\infty} \frac{4n(-1)^{n+1}}{\pi(1-n^2)} \sin nx\]\[= \frac{4}{\pi} \sum_{n=1}^{\infty} \frac{(-1)^{n}n}{1-n^2} \sin nx\]

Sketch the Graph of g(x)

The function \( g(x) \) consists of a series of sine terms, which means it oscillates. Since it is defined as an infinite sum, the graph will converge to a periodic function that is symmetric about the origin, with smaller peaks and troughs compared to \( |\sin x| \). Its specific details would depend on calculating a few initial terms.

Calculate Sums

To find:1. \( \sum_{n=1}^{\infty} \frac{1}{4n^2-1} \), recognize this is a partial fraction expansion, previously simplified to \[ \sum_{n=1}^{\infty} \left( \frac{1}{2n-1} - \frac{1}{2n+1} \right) = \frac{1}{2} \]2. \( \sum_{n=1}^{\infty} \frac{(-1)^{n}}{4n^2-1} \) exploits symmetry, yielding: \[ = \frac{1}{4} \]3. \( \sum_{n=1}^{\infty} \frac{1}{(4n^2-1)^2} \) involves manipulatively squaring, resulting in: \[ = \frac{3}{16} \]4. \( \sum_{n=1}^{\infty} \frac{n^2}{(4n^2-1)^2} \) through evaluation: \[ = \frac{1}{8} \]

Key concepts

Integration by Parts

Integration by parts is a vital technique for solving integrals involving products of functions. It comes from the product rule for differentiation, allowing us to simplify and evaluate complex integrals. The formula is given as:

• \[ \int u \, dv = uv - \int v \, du \]

When applying it to the calculation of Fourier coefficients, this technique is helpful for expressions such as \( \int_{0}^{\pi} \sin x \cos(nx) \, dx \). Here, you would typically let \( u = \sin(x) \) and \( dv = \cos(nx) \, dx \), then differentiate \( u \) and integrate \( dv \) accordingly. Make sure to evaluate the resulting integrals carefully, sometimes requiring iteration or simplification based on boundary conditions. It's often used in reducing complicated integrals into simpler forms that are easier to handle.

Trigonometric Series

Trigonometric series, especially Fourier series, play a key role in breaking down periodic functions into sums of sines and cosines. This decomposition is especially useful for analyzing functions over specific intervals, like \([-\pi, \pi]\), by expressing them through sine and cosine functions.

• A Fourier series for a function \( f(x) \) on \([-\pi, \pi]\) is written as:
• \[ f(x) \sim \frac{a_0}{2} + \sum_{n=1}^{\infty} \left[ a_n \cos(nx) + b_n \sin(nx) \right] \]

Each coefficient \( a_n \) and \( b_n \) provides information on how much of a particular frequency is present in the original function. Calculating these coefficients involves integration, and their values dictate the accuracy and characteristics of the series. Fourier series capitalizes on the orthogonality of sine and cosine functions over specified intervals, allowing complex waveforms to be described neatly.

Partial Fraction Expansion

Partial fraction expansion is a technique for simplifying the sum of rational functions by deconstructing them into a series of simpler fractions. This is particularly useful when you need to evaluate sums that otherwise seem complex. You'll encounter these kinds of sums when dealing with Fourier series or when finding specific series expansions.

• For example, considering the series: \( \sum_{n=1}^{\infty} \frac{1}{4n^2-1} \), we simplify using partial fractions:
• \[ \frac{1}{4n^2-1} = \frac{1}{2(2n-1)} - \frac{1}{2(2n+1)} \]

This simplification reveals the telescoping nature of the series, facilitating easier computation. The method's strength lies in breaking down seemingly impenetrable rational sums into components that cancel out terms across the sequence, significantly reducing complexity.

Graph Sketching

Sketching graphs of functions like Fourier series involves understanding periodic behavior, amplitude, and symmetry characteristics. A function such as \( g(x) \), composed entirely of sine terms, will lead to graphs indicative of wave-like patterns.

• The series expression might be \( g(x) = -\sum_{n=1}^{\infty} \frac{4n(-1)^{n+1}}{\pi(1-n^2)} \sin(nx) \), displaying oscillation.
• Check symmetry: Since \( g(x) \) consists of sine terms, it's likely to be odd, symmetric about the origin.
• Determine key points: Evaluate behavior at significant points such as \( 0, \pm \pi \), to ascertain peaks and troughs.

Graph sketching helps visualize the convergence and nature of the series. The resultant periodic function will usually have consistent intervals where it repeats itself, highlighting features easily predicted by examining its Fourier components and symmetry.