\(\frac{1}{13}+\frac{3}{13.23}+\frac{3}{23.33}+...+\frac{3}{2303.2306}\)
\(=\frac{3}{39}+\frac{3}{13.23}+\frac{3}{23.33}+...+\frac{3}{2303.2306}\)
\(=\frac{3}{13.3}+\frac{3}{13.23}+\frac{3}{23.33}+...+\frac{3}{2303.2306}\)
\(=1-\frac{1}{13}+\frac{1}{13}-\frac{1}{23}+...+\frac{1}{2303}-\frac{1}{2306}\)
\(=1-\frac{1}{2306}=\frac{2305}{2306}\)