Answer
$\sum ^{999}_{k=1}\dfrac {1}{k\times \left( k+1\right) }$
Work Step by Step
$\dfrac {1}{1\times 2}+\dfrac {1}{2\times 3}+\dfrac {1}{3\times 4}+\ldots +\dfrac {1}{999\times 1000}=\dfrac {1}{1\times \left( 1+1\right) }+\dfrac {1}{2\times \left( 2+1\right) }+\dfrac {1}{3\times \left( 3+1\right) }+\ldots +\dfrac {1}{999\times \left( 999+1\right) }=\sum ^{999}_{k=1}\dfrac {1}{k\times \left( k+1\right) }$
