PIL import Image

Load the image to examine its contents

img_path = '/mnt/data/17334635097515cymwgzl.jpg' image = Image.open(img_path)

image.show()It seems that the image is being displayed now, but I need to analyze the contents in order to help you solve the problem. I'll process the image to extract the mathematical details. pytesseract

Use pytesseract to extract text from the image

extracted_text = pytesseract.image_to_string(image)

extracted_textThe problem asks to compute the Fourier Series of the function $f(x) = |x|$ over the interval $-\pi < x < \pi$, and then use the Fourier Series to find an infinite series expansion for $\pi^2$.

Solution Approach

To solve this, we will:

1. Find the Fourier Series of $f(x) = |x|$ on $-\pi < x < \pi$.
2. Use the Fourier coefficients to derive an infinite series expansion for $\pi^2$.

Step 1: Fourier Series for $f(x) = |x|$

The Fourier Series of a function $f(x)$ on the interval $-L < x < L$ is given by:

$f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left( a_n \cos \left( \frac{n\pi x}{L} \right) + b_n \sin \left( \frac{n\pi x}{L} \right) \right)$

Where the coefficients are:

$a_0 = \frac{1}{L} \int_{-L}^{L} f(x) \, dx$ $a_n = \frac{2}{L} \int_{-L}^{L} f(x) \cos \left( \frac{n\pi x}{L} \right) \, dx$ $b_n = \frac{2}{L} \int_{-L}^{L} f(x) \sin \left( \frac{n\pi x}{L} \right) \, dx$

For $f(x) = |x|$ and $L = \pi$, the Fourier Series becomes:

$f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos(nx) + \sum_{n=1}^{\infty} b_n \sin(nx)$

Step 2: Computing the Fourier Coefficients

• $a_0$ (Average value):

  $a_0 = \frac{1}{\pi} \int_{-\pi}^{\pi} |x| \, dx$

  This integral can be split as:

  $a_0 = \frac{1}{\pi} \left( 2 \int_0^{\pi} x \, dx \right) = \frac{2}{\pi} \left[ \frac{x^2}{2} \right]_0^{\pi} = \frac{2}{\pi} \cdot \frac{\pi^2}{2} = \pi$

• $a_n$ (Cosine coefficients):

  $a_n = \frac{2}{\pi} \int_{-\pi}^{\pi} |x| \cos(nx) \, dx$

  Because $|x|$ is an even function and $\cos(nx)$ is also even, the integral simplifies to:

  $a_n = \frac{4}{\pi} \int_0^{\pi} x \cos(nx) \, dx$

  This integral can be solved using integration by parts.

• $b_n$ (Sine coefficients):

  $b_n = \frac{2}{\pi} \int_{-\pi}^{\pi} |x| \sin(nx) \, dx$

  Since $|x|$ is even and $\sin(nx)$ is odd, the integral over the symmetric interval results in zero:

  $b_n = 0$

Step 3: Using the Fourier Series to Find $\pi^2$

After calculating the Fourier coefficients, we can use the series to find an infinite series for $\pi^2$.

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