Solution:

$ \mathrm{x}(t)=\sum_{k=-\infty}^{\infty} C_k \mathrm{e}^{j k \omega_0 t}=\sum_{k=0}^{\infty}\left(-\frac{1}{3}\right)^k \mathrm{e}^{j k \omega_0 t}\\ $

$ \begin{aligned}\\ & =\sum_{k=0}^{\infty}\left(-\frac{1}{3} \mathrm{e}^{j \omega_0 t}\right)^k \\\\ & =1+\left(-\frac{1}{3} \mathrm{e}^{j \omega_0 t}\right)+\left(-\frac{1}{3} \mathrm{e}^{j \omega_0 t}\right)^2 \\\\ & =\frac{1}{\left(1+\frac{1}{3} \mathrm{e}^{j \omega_0 t}\right)} \\\\ & =\frac{1}{\left(1+\frac{1}{3} \mathrm{e}^{j 2 \pi t}\right)}\\ \end{aligned} $

$ \begin{aligned} c_n & =\frac{A}{T} \int_0^d …