= Solution
For $|\phi|<1$, the causal representation is $X_t=\sum_{j\ge0}\phi^j\varepsilon_{t-j}$. Thus
$$
\gamma_k=\frac{\sigma^2}{1-\phi^2}\phi^{|k|}.
$$
Alternatively the rational spectrum from (iii) is $\sigma^2/\{\pi(1-\phi z)(1-\phi z^{-1})\}$ on $|z|=1$. The absolutely convergent geometric sums give
$$
\begin{aligned}
\sum_{k\in\mathbb Z}\phi^{|k|}z^k
&=1+\frac{\phi z}{1-\phi z}+\frac{\phi z^{-1}}{1-\phi z^{-1}}\\
&=\frac{1-\phi^2}{(1-\phi z)(1-\phi z^{-1})}.
\end{aligned}
$$
Consequently
$$
\boxed{f_X(\omega)=\frac{\sigma^2}{\pi(1-\phi^2)}
\sum_{k\in\mathbb Z}\phi^{|k|}e^{ik\omega}.}
$$
This is the <Poisson-kernel expansion of an AR(1) spectrum>, with the same one-sided normalization as (i).
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