Fractions

1+2013=2014

3+2011=2014

5+2009=2014

.........

2013+1=2014

There are $\frac { 1+2013 }{ 2 }$ =1007 pairs of numbers that add up to 2014.

So 1+3+5+...+2013= $\frac { 1007\times 2014 }{ 2 }$ =1007 $\times$ 1007

$\frac { 1007\times 1007 }{ 2014 }$

= $\frac { 1007 }{ 2 }$

=503.5

The numerators are the first 1007 odd numbers so that their sum is $1007^2$ . The total comes out to be $\frac{1007^2}{2014}=\frac{2014}{4}=\boxed{503.5}$

The sum is equivalent to $\frac{1}{2014}\left(\sum_{n=1}^{1007}(2n-1)\right).$ Which when simplified comes out to $\frac{(1007)^2}{2014}=\boxed{503.5}.$