A claim regarding Fourier Series Claim : "A periodic function $f(u)$ satisfying $$\int_{0}^{1}f(u)du=0$$ can generally expanded into a Fourier Series: $$f(u)=\sum_{m=1}^{\infty}[a_m\sin{(2 \pi m u)}+b_m\cos{(2 \pi m u)}]$$ " This is written on Greiner's Classical Mechanics when solving a Tautochrone problem. Firstly,I don’t understand why we didn’t use the term $m=0$ and Sencondly, how the integrand helps us to fulfill the Dirichlet conditions. That means,how do we know that the period is 1?Thanks

If you read Greiner closely, $f(x)$ has period $1$ because that's how it is constructed.
$a_0=0$ because $a_0=\int_{0}^{1}f(u)du=0$