Đặt \(A=\frac15+\frac{1}{13}+\frac{1}{25}+\cdots+\frac{1}{10^2+11^2}\)
\(=\frac15+\frac{1}{13}+\frac{1}{25}+\cdots+\frac{1}{100+121}\)
\(=\frac15+\frac{1}{13}+\frac{1}{25}+\cdots+\frac{1}{221}\)
=>\(A<\frac15+\frac{1}{12}+\frac{1}{24}+\cdots+\frac{1}{220}\)
=>\(A<\frac15+\frac12\left(\frac16+\frac{1}{12}+\cdots+\frac{1}{110}\right)\)
=>\(A<\frac15+\frac12\left(\frac12-\frac13+\frac13-\frac14+\cdots+\frac{1}{10}-\frac{1}{11}\right)\)
=>\(A<\frac15+\frac12\left(\frac12-\frac{1}{11}\right)=\frac15+\frac12\cdot\frac{9}{22}=\frac15+\frac{9}{44}\)
=>\(A<\frac{44}{220}+\frac{45}{220}=\frac{89}{220}\)
=>\(A<\frac{99}{220}=\frac{9}{20}\)
