In the following equation (From https://www.probabilitycourse.com/chapter6/6_1_3_moment_functions.php)
$$ \begin{align}%\label{} \nonumber M_Y(s)&=\frac{e^s-1}{s}\\ \nonumber &=\frac{1}{s} \left(\sum_{k=0}^{\infty} \frac{s^k}{k!}-1\right)\\ \nonumber &=\frac{1}{s} \sum_{k=1}^{\infty} \frac{s^k}{k!}\\ \nonumber &=\sum_{k=1}^{\infty} \frac{s^{k-1}}{k!}\\ \nonumber &=\sum_{k=0}^{\infty}\frac{1}{k+1} \frac{s^{k}}{k!}. \end{align} $$
How the last two line equal?
Maybe do it in two steps \begin{align} \dots&=\sum_{k=1}^{\infty} \frac{s^{k-1}}{k!}\\ &=\sum_{j=0}^{\infty}\frac{1}{j+1} \frac{s^{j}}{j!}\\ &=\sum_{k=0}^{\infty}\frac{1}{k+1} \frac{s^{k}}{k!}. \end{align} For the first step write $k=j+1$ and note $(j+1)! = (j+1)\,j!$.
For the second step, relabel the dummy variable $j$ as $k$.