\(\frac{2}{\sqrt{k+1}+\sqrt{k}}<\frac{2}{2\sqrt{k}}<\frac{2}{\sqrt{k}-\sqrt{k-1}}\)
\(2\left(\sqrt{k+1}-\sqrt{k}\right)<\frac{1}{\sqrt{k}}<2\left(\sqrt{k}-\sqrt{k-1}\right)\)
\(2\sqrt{3}-2\sqrt{2}<\frac{1}{\sqrt{2}}<2\sqrt{2}-2\sqrt{1}\)
\(2\sqrt{4}-2\sqrt{3}<\frac{1}{\sqrt{3}}<2\sqrt{3}-2\sqrt{2}\)
\(2\sqrt{5}-2\sqrt{4}<\frac{1}{\sqrt{4}}<2\sqrt{4}-2\sqrt{3}\)
.......................................................................
\(2\sqrt{101}-2\sqrt{100}<\frac{1}{\sqrt{100}}<2\sqrt{100}-2\sqrt{99}\)
Cộng từng vế ta dc
\(2\sqrt{101}-2\sqrt{2}<\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+....+\frac{1}{\sqrt{100}}<2\sqrt{100}-2\sqrt{1}\)
\(17<\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+....+\frac{1}{\sqrt{100}}<18\)
