Solution
= Solution
Split the transform into periods and translate each interval:
$$
\mathcal L\{g\}(s)=\sum_{j=0}^\infty\int_{jT}^{(j+1)T}e^{-st}g(t)dt
=\sum_{j=0}^\infty e^{-sjT}\int_0^T e^{-su}g(u)du.
$$
Summing the <geometric series> yields
$$
\boxed{\mathcal L\{g\}(s)=\frac{\mathcal L\{g_T\}(s)}{1-e^{-sT}}.}
$$