The value of $$1^3 - 2^3 + 3^3 - ... + 15^3$$ is:

$$S = \sum_{k=1}^{8} (2k-1)^3 - \sum_{k=1}^{7} (2k)^3$$

$$S = \sum_{k=1}^{8} (8k^3 - 12k^2 + 6k - 1) - \sum_{k=1}^{7} 8k^3$$

$$S = \left[ 8\sum_{k=1}^{8} k^3 - 12\sum_{k=1}^{8} k^2 + 6\sum_{k=1}^{8} k - \sum_{k=1}^{8} 1 \right] - 8\sum_{k=1}^{7} k^3$$

$$\sum_{k=1}^{8} k^3 - \sum_{k=1}^{7} k^3 = 8^3 = 512$$

$$S = 8(8^3) - 12\left(\frac{8 \cdot 9 \cdot 17}{6}\right) + 6\left(\frac{8 \cdot 9}{2}\right) - 8$$

$$S = 8(512) - 12(204) + 6(36) - 8$$

$$S = 1856$$