Solution
= Solution
Set $H=\int_0^\pi|h(t)|\,dt<\infty$. By <periodic function>[periodicity] and <countable additivity> of the <Lebesgue integral> of a nonnegative function,
$$
\begin{aligned}
\int_0^\infty e^{-\beta x}|h(x)|\,dx
&=\sum_{k=0}^\infty e^{-\beta k\pi}
\int_0^\pi e^{-\beta t}|h(t)|\,dt\\
&\leq H\sum_{k=0}^\infty e^{-\beta k\pi}
=\frac{H}{1-e^{-\beta\pi}}<\infty.
\end{aligned}
$$
Thus \b[$f$ is Lebesgue integrable] for every $\beta>0$.