Negative frequency
Think about phasors. In trigonometric Fourier series (an and bn) you don't need negative frequencies because the result is given by a sum of sines and cosines.
In exponential Fourier series(cn) the need for negative frequencies yields from the fact that for describing a real function with complex exponentials, it necessary two phasors, one turning clockwise and the other anti clockwise, thus the result is the proyection the phasors on the axis.
Tip: the phasors have same angular rate, same magnitude and opposite phase, that's the trick