To proceed, I'll extract the mathematical content from your image, determine the function $f(x)$, and provide the Fourier series analysis along with detailed steps. Let me process the image for clarity. PIL import Image import pytesseract

Load the image provided by the user

image_path = "/mnt/data/173713995487230vl0crp.jpg" image = Image.open(image_path)

Use OCR to extract text from the image

extracted_text = pytesseract.image_to_string(image) extracted_textThe OCR process provided text, but it appears somewhat cluttered and incomplete. Based on the uploaded image and description, I will reconstruct the relevant part to address the Fourier series of $f(x)$ and show the steps. Let me start with the Fourier series analysis for a generic function:

1. Fourier series for a periodic function $f(x)$ of period $2\pi$ is: $f(x) = a_0 + \sum_{n=1}^\infty \left( a_n \cos(nx) + b_n \sin(nx) \right),$ where: $a_0 = \frac{1}{2\pi} \int_{-\pi}^\pi f(x) \, dx,$ $a_n = \frac{1}{\pi} \int_{-\pi}^\pi f(x) \cos(nx) \, dx,$ $b_n = \frac{1}{\pi} \int_{-\pi}^\pi f(x) \sin(nx) \, dx.$

2. For clarity, I’ll focus on analyzing the Fourier coefficients and summing terms up to $\cos(5x)$ and $\sin(5x)$, followed by sketching partial sums. Could you confirm or clarify the exact function $f(x)$?