$\begin{align*}S_{n+1} &= \sum_{k=1}^{2(n+1)} (-1)^{k+1} k^3 = \sum_{k=1}^{2n} (-1)^{k+1} k^3 + (-1)^{(2n+1)+1} (2n+1)^3 + (-1)^{(2n+2)+1} (2n+2)^3 \\
&= S_n + (-1)^{(2n+1)+1} (2n+1)^3 + (-1)^{(2n+2)+1} (2n+2)^3.
\end{align*}$