\(\frac{1}{1000.1998}+\frac{1}{1001.1997}+...+\frac{1}{1998+1000}\)
\(S=\frac{1}{1000.1998}+\frac{1}{1001.1997}+...+\frac{1}{1998.1000}\)
\(=\frac{1}{2998}\left(\frac{1000+1998}{1000.1998}+\frac{1001+1997}{1001.1997}+...+\frac{1998+1000}{1998.1000}\right)\)
\(=\frac{1}{2998}\left(\frac{1}{1000}+\frac{1}{1998}+\frac{1}{1001}+\frac{1}{1997}+...+\frac{1}{1998}+\frac{1}{1000}\right)\)
\(=\frac{2}{2998}\left(\frac{1}{1000}+\frac{1}{1001}+\frac{1}{1002}+...+\frac{1}{1998}\right)\)
\(=\frac{1}{1499}\left[\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{1998}\right)-\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{199}\right)\right]\)
\(=\frac{1}{1499}\left[\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{1998}\right)-2.\left(\frac{1}{2}+\frac{1}{4}+...+\frac{1}{1998}\right)\right]\)
\(=\frac{1}{1499}\left(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{1997}-\frac{1}{1998}\right)\)
\(=\frac{1}{1499}\left(\frac{1}{1.2}+\frac{1}{3.4}+\frac{1}{5.6}+...+\frac{1}{1997.1998}\right)\)
