Ta có:\(\dfrac{1}{101}+\dfrac{1}{102}+\dfrac{1}{103}+...+\dfrac{1}{200}>\dfrac{1}{200}+\dfrac{1}{200}+\dfrac{1}{200}+...+\dfrac{1}{200}\) (100 số hạng) \(=\dfrac{1}{2}\).
\(\Rightarrow\) đpcm.
\(B=\dfrac{1}{101}+\dfrac{1}{102}+\dfrac{1}{103}+...+\dfrac{1}{200}\)
Vì \(\dfrac{1}{101},\dfrac{1}{102},\dfrac{1}{103},...,\dfrac{1}{199}\)đều lớn hơn \(\dfrac{1}{200}\)
\(\Rightarrow B>\dfrac{1}{200}+\dfrac{1}{200}+\dfrac{1}{200}+...+\dfrac{1}{200}\)(có 100 số hạng \(\dfrac{1}{200}\))
\(\Leftrightarrow B>100\cdot\dfrac{1}{200}\)
\(\Leftrightarrow B>\dfrac{1}{2}\)
Vậy \(B>\dfrac{1}{2}\)
Ta thấy:
\(\dfrac{1}{101}>\dfrac{1}{200}\)
\(\dfrac{1}{102}>\dfrac{1}{200}\)
\(\dfrac{1}{103}>\dfrac{1}{200}\)
...
\(\dfrac{1}{200}=\dfrac{1}{200}\)
=>\(\dfrac{1}{101}+\dfrac{1}{102}+\dfrac{1}{103}+...+\dfrac{1}{109}+\dfrac{1}{200}\)>\(\dfrac{1}{200}+\dfrac{1}{200}+\dfrac{1}{200}+...+\dfrac{1}{200}\)(100 số hạng)=\(\dfrac{1}{2}\)
=> \(\dfrac{1}{101}+\dfrac{1}{102}+\dfrac{1}{103}+...+\dfrac{1}{200}\)>\(\dfrac{1}{2}\)
Vậy B>\(\dfrac{1}{2}\)
