Question regarding the definition of the complex fourier series

[Narcol2000]

Why is the complex Fourier series expanded from $$+\infty$$ to $$-\infty$$.

Yet the regular Fourier series from 0 to $$+\infty$$?

 

[HallsofIvy]

e^inx= cos(nx)+ i sin(nx). cos(x) is an even function while sin(x) is an odd function: cos(-nx)= cos(nx), sin(-nx)= -sin(nx). Negative values of n are alread "incorporated" into the coefficients. e^i(-n)x= cos(-nx)+ isin(-nx)= cos(nx)- i sin(nx) so changing n for -n in the complex exponential gives the complex conjugate.

 

[Narcol2000]

Ah i see that explans the difference of a factor of two between the regular coefficients and complex coefficients...

thanks