Fourier series f(x)=x^2 , (0,2l) also evaluate infinite summation of 1/n^2

Let's work through the problem step by step.

### 1. Fourier Series for \( f(x) = x^2 \) in the Interval \( (0, 2l) \)

The Fourier series of a function \( f(x) \) over the interval \( (0, 2l) \) can be expressed as:

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

For \( f(x) = x^2 \), we need to compute the coefficients \( a_0 \), \( a_n \), and \( b_n \).

#### a) Calculating \( a_0 \)
The coefficient \( a_0 \) is given by:

\[
a_0 = \frac{1}{l} \int_{0}^{2l} f(x) \, dx
\]

Substituting \( f(x) = x^2 \):

\[
a_0 = \frac{1}{l} \int_{0}^{2l} x^2 \, dx = \frac{1}{l} \left[ \frac{x^3}{3} \right]_{0}^{2l} = \frac{1}{l}

Explore the Fourier series of the function f(x) = x^2 over the interval (0, 2l). Learn how to compute Fourier coefficients and understand the mathematical derivation step-by-step. Includes evaluation of an infinite series related to 1/n^2.

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