Solution: Note the factorization $m^2-n^2=(m-n)(m+n)$ is true. Using that on this we get $2013^2-2012^2=1*(2013+2012)$ $2011^2-2010^2=1*(2011+2010)$ and so on. We can prove this pattern will hold by letting $m=x+1$ and $n=x$. The factorization gives us that $(x+1)^2-x^2=x+1+x=2x+1$. Thus the series we were given to evaluate is equal to the sum of the first 2013 positive integers minus 1. So our answer is $2013*2014/2-1=2027090$.