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The Fourier series is a mathematical tool used for analyzing periodic functions by decomposing such a function into a weighted sum of much simpler sinusoidal component

The Fourier series expansion of f is:

$f(t) = rac{1}{2} a_0 + sum_{n=1}^{infty}[a_n cos(omega_n t) + b_n sin(omega_n t)]$

where, for any non-negative integer n:

Let f be periodic of period

2

, with

f (x) = x

for x from -

to

We will compute the Fourier coefficients for this function.

$egin{align} a_n &{}= rac{1}{pi}int_{-pi}^{pi}f(x)cos(nx),dx &{}= rac{1}{pi}int_{-pi}^{pi}x cos(nx),dx &{}= 0. end{align}$

$egin{align} b_n &{}= rac{1}{pi}int_{-pi}^{pi}f(x)sin(nx),dx &{}= rac{1}{pi}int_{-pi}^{pi} x sin(nx), dx &{}= rac{2}{pi}int_{0}^{pi} xsin(nx), dx &{}= rac{2}{pi} left(left[- rac{xcos(nx)}{n} ight]_0^{pi} + left[ rac{sin(nx)}{n^2} ight]_0^{pi} ight) &{}= 2 rac{(-1)^{n+1}}{n}.end{align}$

Notice that a_n are 0 because the $xmapsto xcos(nx)$ are odd functions. Hence the Fourier series for this function is:

$f(x)= rac{a_0}{2} + sum_{n=1}^{infty}left[a_ncosleft(nx ight)+b_nsinleft(nx ight) ight]$

$=2sum_{n=1}^{infty} rac{(-1)^{n+1}}{n} sin(nx), quad orall xin [-pi,pi].$

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