Question

Cmr ∀ n ∈ \(Z^+\)ta luôn có : \(1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+.....+\frac{1}{\sqrt{n}}>2\left(\sqrt{n+1}-1\right)\)

Answer 1

\(A=\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{n}}\)

\(A=2\left(\frac{1}{\sqrt{1}+\sqrt{1}}+\frac{1}{\sqrt{2}+\sqrt{2}}+...+\frac{1}{\sqrt{n}+\sqrt{n}}\right)\)

\(A>2\left(\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+...+\frac{1}{\sqrt{n}+\sqrt{n+1}}\right)\)

\(A>2\left(\sqrt{2}-\sqrt{1}+\sqrt{3}-\sqrt{2}+...+\sqrt{n+1}-\sqrt{n}\right)\)

\(A>2\left(\sqrt{n+1}-1\right)\)