##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:  
 $1.  \cos(2x) = 1 - 2 \sin^2(x)$

$$\begin{array}{ccl} 
e^{iy}  =  1+iy+\frac{(iy)^{2}}{2!}+\frac{(iy)^{3}}{3!}+\frac{(iy)^{4}}{4!}+\frac{(iy)^{5}}{5!}+\cdots\\
  =  1+iy-\frac{y^{2}}{2!}-i\frac{y^{3}}{3!}+\frac{y^{4}}{4!}+i\frac{y^{5}}{5!}+\cdots\\
  =  (1-\frac{y^{2}}{2!}+\frac{y^{4}}{4!}+\cdots)+i(y-\frac{y^{3}}{3!}+\frac{y^{5}}{5!}-\cdots)\\
  =  \cos y+i\sin y\end{array}$$
  
##What is the Fourier series actually?</b>

##Why is Fourier series useful? </b>

$(\nearrow)\cdot(\uparrow)=(\nwarrow)$