Find the value of the sum $\frac{1}{1000*1998}+\frac{1}{1001*1997}+\cdots+\frac{1}{1998*1000}$

Question

Find the value of the sum $\frac{1}{1000*1998}+\frac{1}{1001*1997}+\cdots+\frac{1}{1998*1000}$

I attempted expressing it as a telescoping sum, but I don't know how to. Also, I was curious, is there a closed formula for the general sum $\frac{1}{xy}+\frac{1}{(x+1)(y-1)}+\cdots+\frac{1}{yx}$, for positive integers $x<y$ and if so, how is it derived?

Answer 1

This is $$\sum_{n=1000}^{1998}\frac1{n(2998-n)} =\frac1{2998}\sum_{n=1000}^{1998}\left(\frac1n+\frac1{2998-n}\right) =\frac1{1499}\left(H_{1998}-H_{999}\right)$$ where $H_n$ is the $n$-th harmonic number.