CMR

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

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

\(\frac{\sqrt{2}-\sqrt{1}}{1+2}+\frac{\sqrt{3}-\sqrt{2}}{2+3}+.........+\frac{\sqrt{25}-\sqrt{24}}{24+25}< \frac{2}{5}\)

CMR:

a, \(1+\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{n^2}>\frac{n}{n+1}\)

b, \(2\left(\sqrt{n-1}-1\right)< 1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{n}}< 2\sqrt{n-1}\)

c, \(\frac{1}{2\sqrt{1}}+\frac{1}{3\sqrt{2}}+...+\frac{1}{\left(n+1\right)\sqrt{n}}< 2\)

1) Chứng minh: \(2\sqrt{n}-3< \frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{n}}< 2\sqrt{n}-2\forall n\ge2\)

2) Thu gọn: \(A=5\left(\sqrt{2+\sqrt{3}}+\sqrt{3-\sqrt{5}}-\sqrt{\frac{5}{2}}\right)^2+\left(\sqrt{2-\sqrt{3}}+\sqrt{3+\sqrt{5}}-\sqrt{\frac{3}{2}}\right)^2\)

Chứng minh rằng: \(2\sqrt{n+1}-2\sqrt{n}< \frac{1}{\sqrt{n}}< 2\sqrt{n}-2\sqrt{n-1}\)

Từ đó suy ra: \(2004< 1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{1006009}}< 2005\)

CMR \(2\left(\sqrt{n+1}-\sqrt{n}\right)< \frac{1}{\sqrt{n}}< 2\left(\sqrt{n}-\sqrt{n-1}\right)\)với n thuộc N*

Áp dụng cho S=\(1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{100}}\)

CMR 18<S<19

1. với \(n\in N,n\ge2\). chứng minh \(2\left(\sqrt{n+1}-\sqrt{n}\right)< \frac{1}{\sqrt{n}}< 2\left(\sqrt{n}-\sqrt{n-1}\right)\)

2.chứng minh \(17< \frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\frac{1}{\sqrt{4}}+....+\frac{1}{\sqrt{100}}< 18\)

chứng minh : \(\frac{1}{2\sqrt{2}+1\sqrt{1}}+\frac{1}{3\sqrt{3}+2\sqrt{2}}+...+\frac{1}{\left(n+1\right)\sqrt{n+1}+n\sqrt{n}}< 1-\frac{1}{\sqrt{n+1}}\)