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

Question

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

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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 ?

Answer 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 $$ \frac 1{3026} \ln\left(\frac{1008}{505}\right) $$

Answer 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.

Answer 3

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 $0.0002289$

Answer 4

First $$ \frac1{2k(3026-2k)}=\frac1{4\cdot1513}\left(\frac1k+\frac1{1513-k}\right) $$ So $$ \begin{align} \sum_{k=505}^{1008}\frac1{2k(3026-2k)} &=\frac1{6052}\left(\sum_{k=505}^{1008}\frac1k+\sum_{k=505}^{1008}\frac1{1513-k}\right)\\ &=\frac1{3026}(H_{1008}-H_{504}) \end{align} $$ Since $\lim\limits_{n\to\infty}(H_{2n}-H_n)=\log(2)$, we have the approximation $$ \sum_{k=505}^{1008}\frac1{2k(3026-2k)}\approx\frac{\log(2)}{3026} $$

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Evaluating the values above: $$ \frac1{3026}(H_{1008}-H_{504})=0.000228899998 $$ while $$ \frac{\log(2)}{3026}=0.000229063840 $$

Answer 5

The answers already showed that the expression cannot be more simplified. However, it could be quite accurately approximated since $$S_{m,n}=\sum_{i=m}^n \frac{1}{i}=H_n-H_{m-1}$$ Now, consider that both $m$ and $n$ are large numbers; we can use asymptotic expansions and, at the fourth order, $$S_{m,n}\approx \log \left(\frac{n}{m-1}\right)+\frac{1}{2} \left(\frac{1}{n}-\frac{1}{(m-1)}\right)-\frac{1}{12} \left(\frac{1}{n^2}-\frac{1}{(m-1)^2}\right)+\frac{1}{120} \left(\frac{1}{n^3}-\frac{1}{(m-1)^3}\right)+\cdots$$ Using your numbers, the above approximation gives $0.6926513948612855559$ while the summation leads to $\approx 0.6926513948612855562$