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The questions presented focus on deriving Fourier series expansions for various functions and using them to deduce certain summations. Here's a breakdown:

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**Q1:** Find the Fourier series to represent \( f(x) = x^2 \) in the interval \( (0, \pi) \) and deduce that:

\[
\frac{\pi^2}{12} = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \cdots
\]

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**Q2:** Obtain the Fourier series expansion of \( f(x) = \left( \frac{\pi - x}{2} \right)^2 \) in the interval \( 0 \leq x \leq 2\pi \) and deduce:

1. \[
\frac{\pi^2}{6} = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \cdots
\]
2. \[
\frac{\pi^2}{12} = \frac{1}{1^2} - \frac{1}{2^2} + \frac{1}{3^2} - \frac{1}{4^2} + \cdots
\]

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**Q3:** Find the Fourier series for \( f(x) \) where:

\[
f(x) =
\begin{cases}
-\pi & \text{if } -\pi < x < 0 \\
x & \text{if } 0 < x < \pi
\end{cases}
\]

State the value of \( f(x) \) at \( x = 0 \) and deduce:

\[
\sum_{n=1}^{\infty} \frac{1}{(2n-1)^2} = \frac{\pi^2}{8}
\]

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**Q4:** Find the Fourier series of the function \( f(x) = x \) for \( -\pi \leq x \leq \pi \).

1. Prove that:
\[
\frac{\pi^2}{6} = \sum_{n=1}^{\infty} \frac{1}{n^2}
\]
2. Prove that:
\[
\frac{\pi^2}{12} = \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^2}
\]

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Would you like detailed solutions for these questions, or an explanation of any specific parts?

Here are 5 related questions you might find helpful:

1. How is the Fourier series derived for a piecewise function?
2. What is Parseval's theorem, and how does it relate to deducing summations from Fourier series?
3. What are the conditions under which a Fourier series converges to a given function?
4. Can Fourier series be extended to periodic functions outside of the original interval?
5. How do we calculate Fourier coefficients for non-trivial functions like \( x^2 \)?

**Tip:** Always check the boundary conditions when deriving Fourier series for piecewise functions, as these conditions can affect the behavior of the series at discontinuities.

Find Fourier series for various functions and deduce summation formulas for specific cases.

Explore the Fourier series expansions for functions such as x^2 and piecewise functions, and deduce important summation results, including π²/6 and π²/12, using Parseval's theorem and other related mathematical concepts.

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