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

Chứng minh: \(\frac{1}{3\left(1+\sqrt{2}\right)}+\frac{1}{5\left(\sqrt{2}+\sqrt{3}\right)}+\frac{1}{7\left(\sqrt{3}+\sqrt{4}\right)}+...+\frac{1}{\left(2n+1\right)\left(\sqrt{n}+\sqrt{n+1}\right)}\)\(< \frac{1}{2}\)

chứng minh:

\(\left(1-\frac{2}{6}\right)\left(1-\frac{2}{12}\right)\left(1-\frac{2}{20}\right).....\left(1-\frac{2}{n\left(n+1\right)}\right)>\frac{1}{3\left(n\right)thuộcNsao}\)

CMR với n thuộc N; n>=2 ta có:

\(A=\left(1-\frac{2}{6}\right)\left(1-\frac{2}{12}\right)\left(1-\frac{2}{20}\right)...\left(1-\frac{2}{n\left(n+1\right)}\right)>\frac{1}{3}\)\(\frac{1}{3}\)

Chứng minh với mọi số nguyên dương n, n>=2 ta có :

\(\left(1-\frac{2}{6}\right)\left(1-\frac{2}{12}\right)\left(1-\frac{2}{20}\right)...\left(1-\frac{2}{n\left(n+1\right)}\right)>\frac{1}{3}\)

Chứng minh bất đẳng thức sau: (x≥0;x≠1;x≠\(\frac{1}{4}\)

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

Chứng minh bất đẳng thức sau: (x≥0;x≠1;x≠\(\frac{1}{4}\)

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

Chứng minh bất đẳng thức sau: (x≥0;x≠1;x≠\(\frac{1}{4}\)

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