Given these Fourier coefficients:
$$ X[k]=\begin{cases} 1 & \text{, k even}\\ 2 & \text{, k odd}\\ \end{cases} $$ I want to find the analytical expression for the function. What i tried was using the complex Fourier Series expression : $$ x(t)=\sum_{k=-\infty}^{\infty} X[k]e^{jkw_0t} $$ where $X[k]$ are the Fourier coefficients, j is the imaginary unit , $w_0=\frac {2\pi}T$ and T is the period , in this case $T=4$. Now i'm kind of stuck in this series because i can't find out how to evaluated this sum , what i have so far is ( separated the even and odd terms of the sum and i also assumed $a_o=0$): $$ x(t)=\sum_{k=-\infty}^{\infty} e^{j(2k)kw_0t} + \sum_{k=-\infty}^\infty 2e^{j(2k+1)kw_0t} $$ What should i try next to calculate this sum?