Chứng minh với mọi số nguyên dương, ta có:
\(\frac{1}{\left(n+1\right)\sqrt{n}+n\sqrt{n+1}}=\frac{\left(n+1\right)\sqrt{n}-n\sqrt{n+1}}{\text{[}\left(n+1\right)\sqrt{n}\text{]}^2-\left(n\sqrt{n+1}\right)^2}\)\(=\frac{\left(n+1\right)\sqrt{n}-n\sqrt{n+1}}{\text{ }\left(n+1\right)^2.n-n^2.\left(n+1\right)}=\frac{\left(n+1\right)\sqrt{n}-n\sqrt{n+1}}{\left(n+1\right)n\left(n+1-n\right)}=\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}\)
Áp dụng: Tính B=....
\(=\frac{1}{\sqrt{1}}-\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}+...+\left(\frac{-1}{\sqrt{120}}\right)+\frac{1}{\sqrt{120}}-\frac{1}{\sqrt{121}}=1-\frac{1}{11}=\frac{10}{11}\)
