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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:
$\qquad\qquad\sin^2(x) = ? $
Based on the double angle formula, $\cos(2x) = 1 - 2 \sin^2(x)$
Rearranging,
$\sin^2(x) = \frac{1-\cos(2x)}{2}$
##What is the Fourier series actually?</b>
##Why is Fourier series useful? </b>
$(\nearrow)\cdot(\uparrow)=(\nwarrow)$
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