The question states that $x\left(t\right)$ has Fourier coefficients $a_k=\{x, k=0; j\left(\frac 1 2\right)^{|k|},k\neq0$. I am to determine whether $x\left(t\right)$ is real.
Here is what I've done so far:
For $x\left(t\right)$ to be real, $a_k=a_{-k}^*$ must hold true.
$a_{k,k\neq0}=j\left(\frac12\right)^{|k|}\neq a_{-k,k\neq0}^*=-j\left(\frac12\right)^{|-k|}=-j\left(\frac12\right)^{|k|}$
So, $x\left(t\right)\notin\Bbb R$
Is that correct? I feel like there is some incorrect assumption I made somewhere in there.
Thanks for the help!
Your reasoning is correct, as julien already said. Additional remark: if you factor $j$ out, then the rest is a symmetric series with real coefficients. Therefore, $x$ is $j$ times a real function, i.e., a purely imaginary function.