\frac{1^2-2^2+3^2-...+(-1)^{n-1}\cdot n^2}{(-1)^{n-1}\cdot n^2 }=\frac{-(2^2-1^2)-(4^2-3^2)-...-(n^2-(n-1)^2) }{-n^2 }=\frac{-1\cdot (2+1)-1\cdot (4+3)-...-1\cdot (n+(n-1))}{-n^2 }=\frac{S_{\frac{n}{2}}}{-n^2}