To find the Fourier series of the function $f(x) = |\cos(x)|$ over the interval $0 < x < \pi$, we will compute its Fourier sine series because the function is even and defined on a half-range $0 < x < \pi$.

Steps to compute the Fourier sine series:

1. Fourier Series Form: Since $f(x) = |\cos(x)|$ is even, we focus on the cosine terms in the Fourier series expansion. The general form of the Fourier series of an even function on $(0, \pi)$ is:

   $f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos(nx)$

2. Coefficients: The coefficients $a_n$ are given by:

   $a_0 = \frac{2}{\pi} \int_0^\pi |\cos(x)| \, dx$ $a_n = \frac{2}{\pi} \int_0^\pi |\cos(x)| \cos(nx) \, dx \quad \text{for } n \geq 1$

Calculating the Fourier coefficients:

1. $a_0$ term: To calculate $a_0$:

   $a_0 = \frac{2}{\pi} \int_0^\pi |\cos(x)| \, dx$

   Since $\cos(x)$ is non-negative on $[0, \pi/2]$ and non-positive on $[\pi/2, \pi]$, we can split the integral into two parts:

   $a_0 = \frac{2}{\pi} \left( \int_0^{\pi/2} \cos(x) \, dx - \int_{\pi/2}^\pi \cos(x) \, dx \right)$

   Since the integral of $\cos(x)$ over $[0, \pi]$ is zero, we get:

   $a_0 = \frac{4}{\pi}$

2. $a_n$ terms: Now for $a_n$:

   $a_n = \frac{2}{\pi} \int_0^\pi |\cos(x)| \cos(nx) \, dx$

   Again, split the integral into two regions:

   $a_n = \frac{2}{\pi} \left( \int_0^{\pi/2} \cos(x) \cos(nx) \, dx - \int_{\pi/2}^\pi \cos(x) \cos(nx) \, dx \right)$

   By solving these integrals, the $a_n$ terms can be calculated, but in many cases, the integrals vanish for odd $n$, and for even $n$, the result is non-zero.

Final Result:

The Fourier series of $f(x) = |\cos(x)|$ will look like:

$f(x) = \frac{2}{\pi} + \sum_{n=1, n\, \text{even}}^{\infty} a_n \cos(nx)$

Where $a_n$ values can be computed explicitly.

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Would you like to proceed with detailed calculations of the $a_n$ terms?

Follow-up Questions:

1. What are the specific values for $a_n$ in the series expansion?
2. Can Fourier sine series be used to represent this function instead?
3. What happens when the interval is extended to $[0, 2\pi]$?
4. How does the symmetry of the function affect the Fourier coefficients?
5. Can we use numerical methods to approximate this Fourier series?

Tip: For even functions on symmetric intervals, cosine terms dominate the Fourier series since the sine terms vanish.