Fourier Series Definition Question In several texts, I see that the Fourier series of a function $f=f(t)$ with period $2L$ is defined by
$$Sf(t):=a_0+\sum_{n=1}^{\infty}\left[a_n\cos\frac{n\pi t}{L}+b_n\sin\frac{n\pi t}{L}\right]$$
I wonder why this is not written this way instead:
$$Sf(t):=\sum_{n=0}^{\infty}\left[a_n\cos\frac{n\pi t}{L}+b_n\sin\frac{n\pi t}{L}\right]$$
Thanks in advance for the clarification.
 A: People usually choose to handle the $a_0$ in these two ways (for real Fourier series):
\begin{gather} 
f(t)=\color{blue}{a_0}+\sum_{n=1}^{\infty}\left[a_n\cos\frac{n\pi t}{L}+b_n\sin\frac{n\pi t}{L}\right], \quad -L<t<L,\\
a_0=\color{blue}{{1\over 2L}}\int_{-L}^L f(t)\,dt, \quad
a_n=\color{red}{{1\over L}}\int_{-L}^L f(t)\cos(n\pi t/L)\,dt,\quad
b_n=\color{red}{{1\over L}}\int_{-L}^L f(t)\sin(n\pi t/L)\,dt
\end{gather} 
(which can certainly be written as $\sum_{n=0}^{\infty}\left[a_n\cos\frac{n\pi t}{L}+b_n\sin\frac{n\pi t}{L}\right]$, although in my experience this is not common)
vs.
\begin{gather} f(t)=\color{blue}{{1\over 2}a_0}+\sum_{n=1}^{\infty}\left[a_n\cos\frac{n\pi t}{L}+b_n\sin\frac{n\pi t}{L}\right], \quad -L<t<L,\\
a_0=\color{blue}{{1\over L}}\int_{-L}^L f(t)\,dt, \quad
a_n=\color{red}{{1\over L}}\int_{-L}^L f(t)\cos(n\pi t/L)\,dt,\quad
b_n=\color{red}{{1\over L}}\int_{-L}^L f(t)\sin(n\pi t/L)\,dt
\end{gather} 
They are of course equivalent.
The benefit of the latter formulation is that the factors on the front of the integrals is all the same, but this also requires you to remember the ${1\over 2}$ in front of the $a_0$ in the series.
Hope that helps.
