\(\frac{4}{21}+\frac{4}{77}+\frac{4}{165}+...+\frac{4}{11877}\)
\(=\frac{4}{3.7}+\frac{4}{7.11}+\frac{4}{11.15}+...+\frac{4}{107.111}\)
\(=\frac{1}{3}-\frac{1}{7}+\frac{1}{7}-\frac{1}{11}+\frac{1}{11}-\frac{1}{15}+...+\frac{1}{107}-\frac{1}{111}\)
\(=\frac{1}{3}-\frac{1}{111}\)
\(=\frac{111}{333}-\frac{3}{333}=\frac{108}{333}=\frac{12}{37}\)