Cho \(n\inℕ^∗\) CMR

\(\sqrt{1+\frac{1}{n^2}+\frac{1}{\left(n+1\right)^2}}=1+\frac{1}{n}-\frac{1}{\left(n+1\right)}\)

cho \(n\inℕ\)

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

Bài 1: CMR

   \(\frac{1}{2\sqrt{1}}+\frac{1}{3\sqrt{2}}+\frac{1}{4\sqrt{3}}+........+\frac{1}{\left(n+1\right)\sqrt{n}}>2,n\varepsilonℕ^∗\)

Bài 2: Cho S= \(\frac{1}{3\left(1+\sqrt{2}\right)}+\frac{1}{3\left(\sqrt{2}+\sqrt{3}\right)}+...+\frac{1}{\left(2n+1\right)\left(\sqrt{n}+\sqrt{n+1}\right)}\)

CMR S<\(\frac{1}{2}\)

\(n\ge3;n\inℕ\)

CMR:

\(\frac{1}{a^n\left(b+c\right)}+\frac{1}{b^n\left(c+a\right)}+\frac{1}{c^n\left(a+b\right)}\ge\frac{3}{2}\)

Cho  : 

\(a_{n=\sqrt{1+\left(1+\frac{1}{n}\right)^2}+\sqrt{1+\left(1+\frac{1}{n}\right)^2}}\)

\(\left(n\ge1\right)\)

Đặt :   \(s=\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_{20}}.\)

CM :   \(S\inℕ^∗\)

1, CMR: \(1+\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{n^2}\ge\frac{n}{n+1}\)

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

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

Đặt S_n=\(\frac{1}{3\left(1+\sqrt{2}\right)}+\frac{1}{5\left(\sqrt{2}+\sqrt{3}\right)}+...+\frac{1}{\left(2n+1\right)\left(\sqrt{n}+\sqrt{n+1}\right)}\)

CMR : S_n<\(\frac{1}{2}\)

Đặt S_n= \(\frac{1}{3\left(1+\sqrt{2}\right)}+\frac{1}{5\left(\sqrt{2}+\sqrt{3}\right)}+\frac{...1}{\left(2n+1\right)\left(\sqrt{n}+\sqrt{n+1}\right)}\)

CMR : S_n<\(\frac{1}{2}\)

Chứng minh rằng

\(2\left(\sqrt{n+1}-\sqrt{n}\right)< \frac{1}{\sqrt{n}}< 2\left(\sqrt{n}-\sqrt{n-1}\right)\)với \(n\inℕ^∗\)

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

chứng minh rằng 18<S<19