Question

\(Cho \) \(S=1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{100}}\)

Chứng minh S không phải là số tự nhiên

Answer 1

\(S=\frac{2}{2}+\frac{2}{2\sqrt{2}}+\frac{2}{2\sqrt{3}}+...+\frac{2}{2\sqrt{100}}\)

\(S< 1+\frac{2}{\sqrt{1}+\sqrt{2}}+\frac{2}{\sqrt{2}+\sqrt{3}}+...+\frac{2}{\sqrt{99}+\sqrt{100}}\)

\(S< 1+2\left(\sqrt{2}-\sqrt{1}+\sqrt{3}-\sqrt{2}+...+\sqrt{100}-\sqrt{99}\right)\)

\(S< 1+2\left(\sqrt{100}-\sqrt{1}\right)=19\)

\(S>\frac{2}{\sqrt{1}+\sqrt{2}}+\frac{2}{\sqrt{2}+\sqrt{3}}+...+\frac{2}{\sqrt{100}+\sqrt{101}}\)

\(S>2\left(\sqrt{2}-1+\sqrt{3}-\sqrt{2}+...+\sqrt{101}-\sqrt{100}\right)\)

\(S>2\left(\sqrt{101}-1\right)>2\left(\sqrt{100}-1\right)=18\)

\(\Rightarrow18< S< 19\Rightarrow S\) ko là số tự nhiên