Question

Chứng minh rằng:\(\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+...+\frac{1}{\sqrt{n}}>\sqrt{n}\)

Với \(n\in N;n>1\)

Answer 1

Ta có : \(\frac{1}{\sqrt{1}}>\frac{1}{\sqrt{n}};\frac{1}{\sqrt{2}}>\frac{1}{\sqrt{n}}...;\frac{1}{\sqrt{n}}=\frac{1}{\sqrt{n}}\)

\(\Rightarrow\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+...+\frac{1}{\sqrt{n}}>\frac{1}{\sqrt{n}}+\frac{1}{\sqrt{n}}+...+\frac{1}{\sqrt{n}}\)

\(=n.\frac{1}{\sqrt{n}}=\sqrt{n}\left(dpcm\right)\)