+6251 \(\dfrac{1}{n^2+n}=\dfrac{1}{n(n+1)} = \dfrac{A}{n}+\dfrac{B}{n+1}\\~\\ A+B=0,~A=1,~B=-1\\~\\ \dfrac{1}{n^2+n} = \dfrac{1}{n}-\dfrac{1}{n+1}\)
\(\sum \limits_{n=1}^{1000}~\left(\dfrac 1 n - \dfrac{1}{n+1}\right) = \\~\\ \left(1 - \dfrac 1 2\right) + \left(\dfrac 1 2-\dfrac 1 3\right) + \left(\dfrac 1 3 - \dfrac 1 4\right) + \dots + \left(\dfrac{1}{1000}-\dfrac{1}{1001}\right) = \\~\\ 1-\dfrac{1}{1001} = \dfrac{1000}{1001}\)
.
