Answer
$0$
Work Step by Step
We expand and simplify the sum. We note that the terms alternate from -1 to 1. Since there are an even number of terms, the sum is 0.
$\displaystyle \sum_{n=1}^{20}(-1)^{n}$
$=-1+1-1+1-1+1-1+1-1+1-1+1-1+1-1+1-1+1-1+1$
$=0$