CMR: Với n thuộc N* thì:

\(\frac{1}{\left(n+1\right)\sqrt{n}}< 2\left(\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{x+1}}\right)\)

Từ đó suy ra tổng sau k là số nguyên tố:

\(\frac{1}{2\sqrt{1}}+\frac{1}{3\sqrt{2}}+\frac{1}{4\sqrt{3}}+...+\frac{1}{2008\sqrt{2007}}\)

Các bạn giúp mk với nhé! Mk cần gấp

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

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

bài 1 cmr với mọi số nguyên n ta luôn có 

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

b2 tìm min

\(y=\sqrt{x+2\left(1+\sqrt{x+1}\right)}+\sqrt{x+2\left(1-\sqrt{x+1}\right)}\)

CMR:

Với n thuộc N*

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

CMR:  \(\frac{1}{2\sqrt{1}}+\frac{1}{3\sqrt{2}}+....+\frac{1}{\left(n+1\right)\sqrt{n}}<2\left(\text{Với n là số tự nhiên}\right)\)

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