$\begin{align*}S_1 &= \sum_{k=1}^{2\cdot 2} (-1)^{k+1} k^3 = \sum_{k=1}^4 (-1)^{k+1} k^3 \\ &= (-1)^{1+1} 1^3 + (-1)^{2+1} 2^3 + (-1)^{3+1} 3^3 + (-1)^{4+1} 4^3\end{align*}$