\(B=\frac{1}{17}+\frac{7}{17\cdot27}+\frac{7}{27\cdot37}+...+\frac{7}{1997\cdot2007}\)
\(B=\frac{1}{17}+\frac{7}{10}\left[\frac{10}{17\cdot27}+\frac{10}{27\cdot37}+...+\frac{10}{1997\cdot2007}\right]\)
\(B=\frac{1}{17}+\frac{7}{10}\left[\frac{1}{17}-\frac{1}{27}+\frac{1}{27}-\frac{1}{37}+...+\frac{1}{1997}-\frac{1}{2007}\right]\)
\(B=\frac{1}{17}+\frac{7}{10}\left[\frac{1}{17}-\frac{1}{2007}\right]=\frac{1}{17}+\frac{1393}{34119}=\frac{200}{2007}\)