##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:  

$1.\quad\sin^2(x) =  ?$  
   
Based on the double angle formula,  
  
$\qquad\cos(2x) = 1 - 2 \sin^2(x)$  
  
Rearranging,  
  
$\qquad\sin^2(x) = \frac{1-\cos(2x)}{2}$  
  
$2.\quad\sin(2x).\cos(2x) =  ?$  
   
Based on the double angle formula,  
  
$\qquad\sin(2x) = 2\sin(x)\cos(x)$  
  
Rearranging,
$$\begin{array}{ccl} 
\sin(2x).\cos(x) & = & (2\sin(x)\cos(x))\cos(x)\\
 & = & 1+iy-\frac{y^{2}}{2!}-i\frac{y^{3}}{3!}+\frac{y^{4}}{4!}+i\frac{y^{5}}{5!}+\cdots\\
 & = & (1-\frac{y^{2}}{2!}+\frac{y^{4}}{4!}+\cdots)+i(y-\frac{y^{3}}{3!}+\frac{y^{5}}{5!}-\cdots)\\
 & = & \cos y+i\sin y\end{array}$$
  
$\qquad\sin^2(x) = \frac{1-\cos(2x)}{2}$ 
##What is the Fourier series actually?</b>

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

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