# Math Help - Question about the lower limit of sums in Fourier series

1. ## Question about the lower limit of sums in Fourier series

According to many websites including wikipedia (Fourier series - Wikipedia, the free encyclopedia), the fourier series of f is defined as $\frac{a_0}{2} + \sum_{n=1}^\infty \, [a_n \cos(nx) + b_n \sin(nx)]$.
According to my class notes and wikipedia (Fourier series - Wikipedia, the free encyclopedia), $f(x)=\sum _{n= -\infty} ^{+\infty} \hat f(n)e^{inx}$.
I don't understand why there's a difference of the lower limit of the sums. I'd love to get an explanation.

2. The cosine function is even, and the sine function is odd. Therefore, negative n's are either superfluous (in the cosine case) or can be captured in the b_n's (in the sine case). The exponential function is neither even nor odd. Therefore, to capture all the required information, you must sum over all the integers.

3. Ok so if I understand well, both definitions are equivalent, right?

4. Originally Posted by arbolis
Ok so if I understand well, both definitions are equivalent, right?
Yes, they are equivalent.
You can convert the real Fourier seies into the complex one by writing $\cos nx = \frac12(e^{inx} + e^{-inx})$ and $\sin nx = \frac1{2i}(e^{inx} - e^{-inx})$. Notice that when you do that, you get –n as well as n in the complex exponentials. That is where the negative terms in the complex summation come from.