Chứng minh :

\(1+\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{200}< \dfrac{25}{12}\)

Ta có :

\(1+\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{200}\)

\(=\left(1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}\right)+\left(\dfrac{1}{5}+\dfrac{1}{6}+\dfrac{1}{7}+...+\dfrac{1}{200}\right)\)

\(=\dfrac{25}{12}+\left(\dfrac{1}{5}+\dfrac{1}{6}+\dfrac{1}{7}+...+\dfrac{1}{200}\right)>\dfrac{25}{12}\)

$\mathrm{Vậy\; 1}+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{200}>\frac{25}{12}$