$\begin{align*}S_1 &= \sum_{k=1}^{2\cdot 1} (-1)^{k+1} k^3 = \sum_{k=1}^2 (-1)^{k+1} k^3 \\ &= (-1)^{1+1} 1^3 + (-1)^{2+1} 2^3\end{align*}$