Is it correct to add an initial phase to the Fourier's series?
$$\sum_{k=-\infty}^{+\infty}{X_{k}\exp\left[j\left(2\pi kf_{0}t+\theta_{k}\right)\right]}$$
On my book there is only the following formula:
$$\sum_{k=-\infty}^{+\infty}{X_{k}\exp\left(j2\pi kf_{0}t\right)}$$
EDIT: maybe this writing is more correct?
$$\sum_{k=-\infty}^{+\infty}{X_{k}\exp\left[j\left(2\pi kf_{0}t+k\theta_{k}\right)\right]}$$
If consider:
$$cos\left(2\pi ft+\theta\right)=\frac{A}{2}exp(j1\theta)exp(j2\pi 1ft)+\frac{A}{2}exp(-j1\theta)exp(-j2\pi 1ft)$$
the angle $\theta$ is opposite in the two terms of Fourier's series. In this case the coefficient is $X_{k}exp(jk\theta_{k})$ instead of $X_{k}$.
Thank you for your help.