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##Why Fourier series possible?</b>
We first begin with a few basic identities on the size of sets. Show that the set of possible functions representing sets is not larger than the set of available functions?
##Why Fourier series is plausible?</b>
To show that Fourier series is plausible, let us consider some arbitrary trignometric functions and see if it is possible to express them as the sum of sines and cosines:
$\hspace\sin^2(x) = ?$
$\sin^2(x) = ?$
$$\begin{array}{ccl}
& = & 1+iy-\frac{y^{2}}{2!}-i\frac{y^{3}}{3!}+\frac{y^{4}}{4!}+i\frac{y^{5}}{5!}+\cdots\\
& = & (1-\frac{y^{2}}{2!}+\frac{y^{4}}{4!}+\cdots)+i(y-\frac{y^{3}}{3!}+\frac{y^{5}}{5!}-\cdots)\\
& = & \cos y+i\sin y\end{array}$$
##What is the Fourier series actually?</b>
##Why is Fourier series useful? </b>
$(\nearrow)\cdot(\uparrow)=(\nwarrow)$
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