a) Ta có
$VP = \dfrac{\sqrt{n+1} - \sqrt{n}}{\sqrt{n(n+1)}}$
$= \dfrac{n + 1 - n}{\sqrt{n(n+1)} (\sqrt{n+1} + \sqrt{n})}$
$= \dfrac{1}{(n+1)\sqrt{n} + n \sqrt{n+1}} = VT$
b) Ta có
$u_n = \dfrac{1}{1 \sqrt{2} + 2\sqrt{1}} + \cdots + \dfrac{1}{(n+1)\sqrt{n} + n \sqrt{n+1}}$
Áp dụng Câu a) ta có
$u_n = \dfrac{1}{1} - \dfrac{1}{\sqrt{2}} + \dfrac{1}{\sqrt{2}} - \dfrac{1}{\sqrt{3}} + \cdots + \dfrac{1}{\sqrt{n}} - \dfrac{1}{\sqrt{n+1}}$
$= 1 - \dfrac{1}{\sqrt{n+1}}$
c) Ta có
$\underset{n \to +\infty}{\lim} u_n = \underset{n \to +\infty}{\lim} 1 - \dfrac{1}{\sqrt{n+1}}$
$= 1 - \underset{n \to +\infty}{\lim} \dfrac{1}{\sqrt{n+1}}$
$= 1 - \underset{n \to +\infty}{\lim} \dfrac{1}{\sqrt{n}} . \dfrac{1}{\sqrt{1 + \frac{1}{n}}}$
$= 1 - 0 = 1$
