Cần giúp đỡ chứng minh BDT

[dieuuocsaobang]

CHỨNG MINH BẤT ĐẲNG THỨC (BDT)​

Bài 1 \[\frac{1}{2\sqrt{1}}+\frac{1}{3\sqrt{2}}+...+\frac{1}{(n+1)\sqrt{n}}<2\] mọi n
Bài 2 \[\sum_{n}^{k=1}\frac{1}{(2k+1)(2k-1)}<\frac{1}{2}\] mọi n

 

[binhdt]

Bài 1 \[\frac{1}{2\sqrt{1}}+\frac{1}{3\sqrt{2}}+...+\frac{1}{(n+1)\sqrt{n}}<2\] mọi n

\[\frac{1}{(n+1)\sqrt{n}} = \frac{\sqrt{n}}{n(n+1)} =\sqrt{n}(\frac{1}{n}-\frac{1}{n+1}) =\sqrt{n}(\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}})(\frac{1}{\sqrt{n}}+\frac{1}{\sqrt{n+1}})<2(\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}})\]
\[=>VT< 2(1-\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}-....+ \frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}})<2\]