Ta có :
\(A=\left(\frac{1}{11}+\frac{1}{12}+...+\frac{1}{20}\right)+\left(\frac{1}{21}+\frac{1}{22}+...+\frac{1}{30}\right)\)
Ta thấy :
\(\frac{1}{20}< \frac{1}{11}\)
\(\frac{1}{20}< \frac{1}{12}\)
\(...\)
\(\frac{1}{20}< \frac{1}{19}\)
\(\Rightarrow\frac{1}{20}\cdot10< \frac{1}{11}+\frac{1}{12}+...+\frac{1}{20}\)
\(\Rightarrow\frac{1}{2}< \frac{1}{11}+\frac{1}{12}+...+\frac{1}{20}\)(1)
\(\frac{1}{30}< \frac{1}{21}\)
\(\frac{1}{30}< \frac{1}{22}\)
\(...\)
\(\frac{1}{30}< \frac{1}{29}\)
\(\Rightarrow\frac{1}{30}\cdot10< \frac{1}{21}+\frac{1}{22}+...+\frac{1}{30}\)
\(\Rightarrow\frac{1}{3}< \frac{1}{21}+\frac{1}{22}+...+\frac{1}{30}\)(2)
Từ (1),(2) :
\(\Rightarrow\frac{1}{2}+\frac{1}{3}< \frac{1}{11}+\frac{1}{12}+...+\frac{1}{30}\)
\(\Rightarrow\frac{5}{6}< A\)
\(#Louis\)
