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

To find the Fourier series for the function \( f(x) = x^2 \) over the interval \( (0, 2L) \) and evaluate the infinite summation \( \sum_{n=1}^{\infty} \frac{1}{n^2} \), let's proceed step by step.

### 1. Fourier Series for \( f(x) = x^2 \) over \( (0, 2L) \):

The Fourier series for a function \( f(x) \) over the interval \( (0, 2L) \) 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 \( a_0 \), \( a_n \), and \( b_n \) are given by:

\[
a_0 = \frac{1}{L} \int_{0}^{2L} f(x) \, dx
\]
\[
a_n = \frac{1}{L} \int_{0}^{2L} f(x) \cos\left(\frac{n\pi x}{L}\right) \, dx
\]
\[
b_n = \frac{1}{L} \int_{0}^{2L} f(x) \sin\left(\frac{n\pi x}{L}\right) \, dx
\]

Since \( f(x) = x^2 \) is an even function (symmetric about the y-axis), all the sine terms (\( b_n \)) will be zero. So we only need to calculate \( a_0 \) and \( a_n \).

#### Calculate \( a_0 \):
\[
a_0 = \frac{1}{L} \int_{0}^{2L} x^2 \, dx = \frac{1}{L} \left[ \frac{x^3}{3} \right]_{0}^{2L} = \frac{1}{L} \times \frac{(2L)^3}{3} = \frac{8L^2}{3}
\]

#### Calculate \( a_n \):
\[
a_n = \frac{1}{L} \int_{0}^{2L} x^2 \cos\left(\frac{n\pi x}{L}\right) \, dx
\]

This integral can be solved using integration by parts, but the detailed calculation involves some steps. The result for \( a_n \) is:

\[
a_n = \frac{4L^2(-1)^n}{n^2 \pi^2}
\]

#### Final Fourier Series:
Putting it all together, the Fourier series for \( f(x) = x^2 \) over \( (0, 2L) \) is:

\[
f(x) = \frac{4L^2}{3} + \sum_{n=1}^{\infty} \frac{4L^2(-1)^n}{n^2 \pi^2} \cos\left(\frac{n\pi x}{L}\right)
\]

### 2. Evaluate the Infinite Summation \( \sum_{n=1}^{\infty} \frac{1}{n^2} \):

This is a well-known result in mathematics. The summation is given by:

\[
\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}
\]

This result comes from the Basel problem, solved by Euler in the 18th century.

### Summary:
- The Fourier series for \( f(x) = x^2 \) over \( (0, 2L) \) is:

\[
f(x) = \frac{4L^2}{3} + \sum_{n=1}^{\infty} \frac{4L^2(-1)^n}{n^2 \pi^2} \cos\left(\frac{n\pi x}{L}\right)
\]

- The infinite summation \( \sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6} \).

Would you like more details on any part of this solution, or do you have any further questions?

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**Relative Questions:**

1. How do you perform integration by parts in calculating \( a_n \)?
2. What is the significance of the Fourier series in signal processing?
3. Can the Fourier series be applied to other functions, and how does the result change?
4. What are the boundary conditions for applying the Fourier series?
5. How does the periodic extension of \( f(x) = x^2 \) look like?
6. What is the importance of the Basel problem in mathematics?
7. Can you find the Fourier series for a different interval, say \( (-L, L) \)?
8. How do odd functions affect the Fourier coefficients?

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**Tip:** When calculating Fourier series, always check the symmetry of the function. This can simplify the calculation by eliminating certain terms (like sine or cosine terms).

Explore the Fourier series for the function f(x) = x^2 over the interval (0, 2L) and evaluate the infinite summation ∑(1/n^2). Understand the derivation of Fourier coefficients and the solution to the Basel problem. Suitable for advanced undergraduate students and enthusiasts of mathematical analysis.

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