Tổng quát:$ \frac{1}{(n+1)\sqrt{n}}$ = $\frac{\sqrt{n}}{(n+1)n} $ = $\sqrt{n}. \frac{1}{n(n+1)}$
=$\sqrt{n}.(\frac{1}{n} - \frac{1}{n+1})$
=$\sqrt{n}.(\frac{1}{(\sqrt{n})^{2}}-\frac{1}{(\sqrt{n+1)}^{2}}$)
=$\sqrt{n}.(\frac{1}{\sqrt{n}}+\frac{1}{\sqrt{n+1}})(\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}$)
= $(1+\frac{\sqrt{n}}{\sqrt{n+1}})(\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}})$
(Do $\frac{\sqrt{n}}{\sqrt{n+1}} = \sqrt{\frac{n}{n+1}} < \sqrt{\frac{n+1}{n+1}} = \sqrt{1} = 1$)
< (1+1)($\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}})$
< 2($\frac{1}{\sqrt{n}}- \frac{1}{\sqrt{n+1}})$
Cho n=1;2;3;...;2014. Ta có: (Gọi A là biểu thức cần CM)
A < 2 (1 - $\frac{1}{\sqrt{2015}}$) = 2 - $\frac{2}{\sqrt{2015}}$ < 2
Vậy A < 2