1/1×3+1/3×5+1/5×7+...+1/2013×2015
=2×(1/1×3+1/3×5+1/5×7+...+1/2013×2015)÷2
=(2/1×3+2/3×5+2/5×7+...+2/2013×2015)÷2
=(1-1/3+1/3-1/5+1/5-1/7+1/7-1/9+.+1/2011-1/2013+1/2013-1/2015)÷2
=(1-1/2015)÷2
=2014/2015÷2
=1007/2015