Question

Consider the function \(f(x)=x,-\pi

Short answer

The given expression in part (a) is a Fourier sine series. After integrating the series, as required in part (b), the result matches with the Fourier cosine series of \(f(x)=x^{2}\) on \((0, \pi)\), computed in part (c); thus validating the computations.

Step-by-step solution

Show the Fourier series

Start by writing the left-hand side of the equation and carefully compute the given sum. The right-hand side involves a sine series. Each sine term is multiplied by \(-1^{n+1}\)/n. It's a classic Fourier sine series.

Integrate the series

First, integrate the left-hand side to get \(x^2 \)/2. Then, integrate each term of the series on then right-hand side. Use the property that the integral of \( \sin(nx)\) is \(-\cos(nx)/n\). Do not forget to apply the constant of integration which will be found by setting \(x = 0\). Compare the two expressions and simplify to find the constant.

Find the Fourier cosine series

To find the Fourier cosine series of \(f(x)=x^{2}\) on \((0, \pi)\), recall the formula for the Fourier cosine series: \(f(x)=a_{0}/2 + \sum_{n=1}^\infty[a_{n}\cos(nx)]\) where \(a_{n}=(2/\pi)\int_{0}^{\pi}f(x)\cos(nx)dx\). Substitute \(f(x)=x^{2}\) into the formula and calculate the integrals.

Compare the results

Compare the series obtained in parts (b) and (c). They should match, confirming that the calculations in previous steps are correct.

Key concepts

Understanding Fourier Sine Series

The Fourier sine series is a way to represent an odd function as an infinite sum of sine functions. Essentially, any periodic function can be broken down into simpler trigonometric components. For the function \(f(x) = x\) in the interval \(-\pi < x < \pi\), the Fourier sine series comes into play. The series is written as:

• Each term involves a sine function multiplied by \((-1)^{n+1}/n\).
• This alternating sign pattern (\((-1)^{n+1}\)) means each sine wave contributes directionally, either adding or subtracting at a given point.
• The term \(1/n\) controls the amplitude of each sine wave, making higher frequency components smaller.

If you write \(x\) as:\[x = 2 \sum_{n=1}^{\infty}(-1)^{n+1} \frac{\sin n x}{n},\]you're expressing \(x\) as an infinite sine series, capturing the essence of the function using an odd extension.

Exploring Fourier Cosine Series

The Fourier cosine series represents functions over a symmetric interval where the focus is on cosine functions. For \(f(x) = x^2\) defined on \((0, \pi)\), the series is a sum of cosine terms:

• The cosine series is constructed using the formula \(f(x) = a_{0}/2 + \sum_{n=1}^\infty[a_{n}\cos(nx)]\).
• Here, \(a_{n}\) are coefficients found through integration: \((2/\pi)\int_{0}^{\pi}f(x)\cos(nx)dx\).

The coefficients \(a_{n}\) ensure each cosine term accurately contributes to forming the function. This creates a complete representation that aligns with the function \(x^2\) over the interval \(0\) to \(\pi\). By comparing the cosine series with part (b)'s result, we verify accuracy and consistency in expressing the function.

Integration of Series

Integration of a series involves summing integrals of individual terms of the series. It provides insight into how the function behaves when extended into different forms. Starting with:

• Integrating \(x = 2 \sum_{n=1}^{\infty}(-1)^{n+1} \frac{\sin n x}{n}\) gives the left as \(x^2/2\).
• Each term transforms, where the integral of \(\sin(nx)\) becomes \(-\cos(nx)/n\), factoring in each component's nature.
• A constant of integration is identified by evaluating the function at a known point, like \(x = 0\), ensuring equilibrium between the series and the new function form.

This process eventually reveals: \[x^{2}=\frac{\pi^{2}}{3}-4 \sum_{n=1}^{\infty}(-1)^{n+1} \frac{\cos n x}{n^{2}}.\] It ties integration with series expansion, illustrating advanced manipulation of function representations.