Lời giải:
$S=\frac{\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{1000.1001.1002}}{sin(2)+sin(4)+sin(6)+...+sin(1000000)}$
Ta có:
$\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{1000.1001.1002}$
$=\frac{1000.1003}{4.1001.1002}=\frac{1003000}{4012008}$
$sin(2)+sin(4)+sin(6)+...+sin(1000000)$
$=sin(2)+sin(2.2)+sin(3.2)+...+sin(500000.2)$
$=\frac{sin\frac{500000.2}{2}.sin\frac{(500000+1).2}{2}}{sin\frac{2}{2}}$
$=\frac{sin(500000).sin(500001)}{sin(1)}$
$=>S=\frac{1003000.sin(1)}{4012008.sin(500000).sin(500001)}$