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