Consider a power series (complex in the general case): $$\sum_{n \geq 0} \mathrm{a_n (z-z_0)^n}$$ It converges at least for $\mathrm{z=z_0}$ and the sum function $F(z)$ in $z_0$ is $a_0$, that is
$$\mathrm{F(z)}=\sum_{n \geq 0} \mathrm{a_n (z-z_0)^n \,\,\,\, and \,\,\,\, F(z_0)=a_0}$$
Why is $F(z_0)=a_0$? Shouldn't is be equal to $0$ since the series in $z=z_0$ is the series $\sum_{n \geq 0} 0$ ?
No, because $\lim_{x\rightarrow 0} x^0=1$. The first term, the $n=0$ term, is nonzero.