Fourier Series Proof
You just need to split the sum into the positive and the negative indices (without forgetting the index $k=0$):

$$x(t)=a_0+\sum_{k=1}^{\infty}a_ke^{jk\omega_0 t}+\sum_{k=-\infty}^{-1}a_ke^{jk\omega_0 t}\tag{1}$$

By simply changing the sign of the index $k$ in the second sum in $(1)$ you get

$$\begin{align}x(t)&amp;=a_0+\sum_{k=1}^{\infty}a_ke^{jk\omega_0 t}+\sum_{k=1}^{\infty}a_{-k}e^{-jk\omega_0 t}\\&amp;=
a_0+\sum_{k=1}^{\infty}\left[a_ke^{jk\omega_0 t}+a_{-k}e^{-jk\omega_0 t}\right]\end{align}\tag{2}$$