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

Solution 1:

You are almost certainly on the correct road. We can rewrite this sum as $$ \frac{1}{6052} \cdot \left(\sum_{n = 0}^{503} \underbrace{\frac 1{n+505}}_{i = n+505} + \sum_{n = 0}^{503} \underbrace{\frac 1{1008-n}}_{j = 1008-n}\right) =\\ \frac{1}{6052} \cdot \left(\sum_{i = 505}^{1008} \frac 1{i} + \sum_{j = 505}^{1008} \frac 1{j}\right) = \frac{1}{3026}\sum_{i = 505}^{1008} \frac 1{i} $$ This doesn't simplify nicely, but it is well approximated by $$ \frac 1{3026} \ln\left(\frac{1008}{505}\right) $$

Solution 2:

The answer is $$\frac{H_{1008}-H_{504}}{3026},$$ where $H_n$'s denote harmonic numbers. I don't see, however, how this can be simplified further.