Why does the Fourier series not include phase information?
(The following is the answer to the original version of the question asking about the Fourier Transform. But exactly the same argument also applies to Fourier series).

They are. Assume that $F(\omega)$ is the Fourier transform of the signal $f(t)$:

$$F(\omega)=\int_{-\infty}^{\infty}f(t)e^{-j\omega t}dt$$

In general, $F(\omega)$ is a complex-valued function and can be written in polar form

$$F(\omega)=|F(\omega)|e^{j\phi(\omega)}$$

where $\phi(\omega)$ is the phase of $F(\omega)$.
The signal $f(t)$ can be written in terms of $F(\omega)$ by virtue of the inverse Fourier transform:

$$f(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}F(\omega)e^{j\omega t}d\omega=
\frac{1}{2\pi}\int_{-\infty}^{\infty}|F(\omega)|e^{j(\omega t+\phi(\omega))}d\omega\tag{1}$$

From (1) it can be seen that $f(t)$ is represented by complex sinusoids with varying amplitudes $|F(\omega)|$, phases $\phi(\omega)$, and frequencies $\omega$.