What is  $(-1)^1+(-1)^2+\cdots+(-1)^{2006}$ ?
I notice that this is a sum of powers of -1, which alternate between -1 and 1 depending on whether the exponent is odd or even. So I can group the terms into pairs of consecutive powers, such as $(-1)^1+(-1)^2$, $(-1)^3+(-1)^4$, and so on. Each pair has an odd power and an even power, so they add up to zero. There are 1003 such pairs, since 2006 divided by 2 is 1003. So the sum is equal to zero times 1003, which is zero. 