\(S=\frac{1}{4}\left(1+\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{50^2}\right)\)
\(S< \frac{1}{4}\left(1+\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{49.50}\right)\)
\(S< \frac{1}{4}\left(1+1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{49}-\frac{1}{50}\right)\)
\(S< \frac{1}{4}\left(1+1-\frac{1}{50}\right)\)
\(S< \frac{1}{4}.\frac{99}{50}=\frac{99}{200}< \frac{1}{2}\)
VẬY\(S< \frac{1}{2}\)