Sửa: \(\dfrac{1}{\sqrt{2}}+\dfrac{1}{\sqrt{3}}+\dfrac{1}{\sqrt{4}}+...+\dfrac{1}{\sqrt{x}}< 2\left(\sqrt{x}-1\right)\)
Cần cm: \(\dfrac{1}{\sqrt{k}}< \dfrac{2}{\sqrt{k}+\sqrt{k-1}}\left(k\in N\text{*},k\ge2\right)\)
\(\Leftrightarrow\sqrt{k}+\sqrt{k-1}< 2\sqrt{k}\\ \Leftrightarrow\sqrt{k}>\sqrt{k-1}\\ \Leftrightarrow k>k-1\left(luôn.đúng\right)\)
Áp dụng: \(\dfrac{1}{\sqrt{2}}< \dfrac{2}{\sqrt{2}+\sqrt{1}};...;\dfrac{1}{\sqrt{x}}< \dfrac{2}{\sqrt{x}+\sqrt{x-1}}\)
\(\Leftrightarrow\dfrac{1}{\sqrt{2}}+\dfrac{1}{\sqrt{3}}+...+\dfrac{1}{\sqrt{x}}< \dfrac{2}{\sqrt{2}+\sqrt{1}}+\dfrac{2}{\sqrt{3}+\sqrt{2}}+...+\dfrac{2}{\sqrt{x}+\sqrt{x-1}}\\ \Leftrightarrow\dfrac{1}{\sqrt{2}}+\dfrac{1}{\sqrt{3}}+...+\dfrac{1}{\sqrt{x}}< 2\left(\sqrt{2}-\sqrt{1}\right)+2\left(\sqrt{3}-\sqrt{2}\right)+...+2\left(\sqrt{x}-\sqrt{x-1}\right)\\ \Leftrightarrow\dfrac{1}{\sqrt{2}}+\dfrac{1}{\sqrt{3}}+...+\dfrac{1}{\sqrt{x}}< 2\sqrt{2}-2+2\sqrt{3}-2\sqrt{2}+...+2\sqrt{x}-2\sqrt{x-1}\\ \Leftrightarrow\dfrac{1}{\sqrt{2}}+\dfrac{1}{\sqrt{3}}+...+\dfrac{1}{\sqrt{x}}< 2\sqrt{x}-2=2\left(\sqrt{x}-1\right)\)
