Factor of The Sequence

Relevant wiki: Sum of n, n², or n³

$\begin{aligned} S & = -1^3 + 2^3 - 3^3 + 4^3 - \cdots - 2013^3 + 2014^3 \\ & = - \left(1^3 + 2^3 + 3^3 + \cdots + 2014^3 \right) + 2 \left( 2^3 + 4^3 + 6^3 + \cdots + 2014^3 \right) \\ & = - \left(1^3 + 2^3 + 3^3 + \cdots + 2014^3 \right) + 16 \left( 1^3 + 2^3 + 3^3 + \cdots + 1007^3 \right) \\ & = - \left(\frac {2014(2015)}2\right)^2 + 16 \left(\frac {1007(1008)}2\right)^2 \\ & = - 1007^2\times 2015^2 + 4\times 1007^2 \times 1008^2 \\ & = 1007^2 \left(2016^2-2015^2 \right) \\ & = 1007^2 ( 2016-2015 ) ( 2016+2015) \\ & = 1007^2 \times 4031 \\ & = 1007^2 \times 29 \times 139 \end{aligned}$

Therefore, the largest perfect square that divides $S$ is $\boxed{1007^2}$ .

Using the summation for cubes (n^2 (n+1)^2)/4...since the signs alternate, we use (-1)^(n+1) to match our signs.

From there plugging in n = 2014 becomes the following:

(-1)^(2015)( (2014)^2 (2015)^2)/4

2014 = 1007 * 2

We don't really care about the negative sign as it has nothing to do with what were looking for,but I'll leave it in , so the sum becomes (-1)((1007*2)^2 (2015)^2)/4

=(-1) (1007)^2 (2015)^2

Hence (1007)^2