Find the Fourier series of f(x) = x+ x2 in (−π,π) of periodicity 2π. Hence deduce
that 1
π2
=
n2
6.

We are tasked with finding the Fourier series of \( f(x) = x + x^2 \) over the interval \( (-\pi, \pi) \) with periodicity \( 2\pi \), and deducing that \( \frac{1}{\pi^2} = \frac{1}{6} \sum_{n=1}^{\infty} \frac{1}{n^2} \).

### Step 1: Fourier Series Expansion
For a \( 2\pi \)-periodic function, the Fourier series is given by:

\[
f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos(nx) + \sum_{n=1}^{\infty} b_n \sin(nx)
\]
where the Fourier coefficients \( a_0 \), \( a_n \), and \( b_n \) are calculated as follows:

\[
a_0 = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \, dx
\]
\[
a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos(nx) \, dx
\]
\[
b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin(nx) \, dx


Find the Fourier series of f(x) = x + x^2 in (−π, π) of periodicity 2π. Hence deduce that 1/π^2 = Σ(1/n^2) / 6.

Find the Fourier series of the function f(x) = x + x^2 over the interval (−π, π) with periodicity 2π. This detailed solution involves calculating Fourier coefficients and using Parseval's Theorem to deduce the famous Basel Problem result: Σ(1/n^2) = π^2/6.

Math Problem Statement

Find the Fourier series of f(x) = x+ x2 in (−π,π) of periodicity 2π. Hence deduce that 1 π2

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