According to many websites including wikipedia (Fourier series - Wikipedia, the free encyclopedia), the fourier series of f is defined as $\displaystyle \frac{a_0}{2} + \sum_{n=1}^\infty \, [a_n \cos(nx) + b_n \sin(nx)]$.

According to my class notes and wikipedia (Fourier series - Wikipedia, the free encyclopedia), $\displaystyle f(x)=\sum _{n= -\infty} ^{+\infty} \hat f(n)e^{inx}$.

I don't understand why there's a difference of the lower limit of the sums. I'd love to get an explanation.