Đặt \(A=\frac15+\frac{1}{13}+\frac{1}{25}+\cdots+\frac{1}{10^2+11^2}\)

\(=\frac15+\frac{1}{13}+\frac{1}{25}+\cdots+\frac{1}{100+121}\)

\(=\frac15+\frac{1}{13}+\frac{1}{25}+\cdots+\frac{1}{221}\)

=>\(A<\frac15+\frac{1}{12}+\frac{1}{24}+\cdots+\frac{1}{220}\)

=>\(A<\frac15+\frac12\left(\frac16+\frac{1}{12}+\cdots+\frac{1}{110}\right)\)

=>\(A<\frac15+\frac12\left(\frac12-\frac13+\frac13-\frac14+\cdots+\frac{1}{10}-\frac{1}{11}\right)\)

=>\(A<\frac15+\frac12\left(\frac12-\frac{1}{11}\right)=\frac15+\frac12\cdot\frac{9}{22}=\frac15+\frac{9}{44}\)

=>\(A<\frac{44}{220}+\frac{45}{220}=\frac{89}{220}\)

=>\(A<\frac{99}{220}=\frac{9}{20}\)

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đơn giải thôi nhưng mình ko bấn fx đc

​

xét vế trái : \(\frac{1}{5}+\frac{1}{13}+\frac{1}{25}+...+\frac{1}{221}\)

ta có : \(T< \frac{1}{5}+\frac{1}{12}+\frac{1}{24}+...+\frac{1}{220}\)

             \(=\frac{1}{5}+\frac{1}{2}\left(\frac{1}{6}+\frac{1}{12}+...+\frac{1}{110}\right)=\frac{1}{5}+\frac{1}{2}\left(\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{10.11}\right)\)

             \(=\frac{1}{5}+\frac{1}{2}\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{10}-\frac{1}{11}\right)\)

              \(=\frac{1}{5}+\frac{1}{2}\left(\frac{1}{2}-\frac{1}{11}\right)< \frac{1}{5}+\frac{1}{4}\Rightarrow T< \frac{9}{20}\)

\(A=\frac{10}{27}+\frac{9}{16}\frac{11}{34}\)

Ta có: \(\frac{10}{27}< >\backslash\left(\frac{9}{16}< >\backslash\left(\frac{11}{34}< >Nên\backslash\left(A< >b\right)\right)\right)\backslash\left(B=\frac{1}{12}+\frac{1}{13}+\frac{1}{14}+...+\frac{1}{22}\right)\)

\(B>\frac{1}{22}+\frac{1}{22}+\frac{1}{22}+...+\frac{1}{22}=11.\frac{1}{22}=\frac{1}{2}\)

Nên \(B>\frac{1}{2}\)

Nhân 2 vế với 5 suy ra 
1+1/3+1/5+...+1/397<9/4 hay 
1/3+1/5+...+1/397<5/4 
Trong khi tính thử bằng máy tính thì 
1/3+1/5+...+1/397>1/3+1/5+1/7+...+1/25 = 1,2643528....>5/4

tích nha