\(\left(-1\right)+\left(-1\right)^2+\left(-1\right)^3+...+\left(-1\right)^{2015}\)
\(=\left[\left(-1\right)+\left(-1\right)^2\right]+\left[\left(-1\right)^3+\left(-1\right)^4\right]+...+\left[\left(-1\right)^{2013}+\left(-1\right)^{2014}\right]+\left(-1\right)^{2015}\)
\(=\left(-1+1\right)+\left(-1+1\right)+....+\left(-1+1\right)-1=-1\)