Evaluate $\frac{ 1 }{ 1010 \times 2016} + \frac{ 1 }{ 1012 \times 2014} + \frac{ 1 }{ 1014 \times 2012} + \cdots + \frac{ 1 }{ 2016 \times 1010} = ?$ $$\dfrac{ 1 }{ 1010 \times 2016} + \dfrac{ 1 }{ 1012 \times 2014} + \dfrac{ 1 }{ 1014 \times 2012} + \cdots + \dfrac{ 1 }{ 2016 \times 1010} = ?
$$

My attempt so far :
$$\sum\limits_{n=0}^{503}\dfrac{1}{(1010+2n)(2016-2n)} =  \dfrac{1}{6052}\sum\limits_{n=0}^{503}\left(\dfrac{1}{n+505} - \dfrac{1}{n-1008}\right)$$
It won't telescope/simplify further. I feel I am in wrong road. Any help ?

First, note that each term is duplicated  $$\dfrac{ 1 }{ 1010 \times 2016} + \dfrac{ 1 }{ 1012 \times 2014} + \dfrac{ 1 }{ 1014 \times 2012} + \cdots + \dfrac{ 1 }{ 2016 \times 1010} = \\\dfrac{2}{ 1010 \times 2016}+  \dfrac{ 2 }{ 1012 \times 2014} + \dfrac{ 2 }{ 1014 \times 2012} + \cdots + \dfrac{ 2 }{ 1512 \times 1514}=\\\sum_{i=0}^{251}\frac 2{1513^2-(2i+1)^2}$$  Alpha gets a fraction with huge numbers that is about $0.0002289$
