\(S=\frac{2}{2}+\frac{2}{2\sqrt{2}}+\frac{2}{2\sqrt{3}}+...+\frac{2}{\sqrt{100}}\)
\(S>\frac{2}{\sqrt{1}+\sqrt{2}}+\frac{2}{\sqrt{2}+\sqrt{3}}+\frac{2}{\sqrt{3}+\sqrt{4}}+...+\frac{2}{\sqrt{100}+\sqrt{101}}\)
\(S>2\left(\sqrt{2}-\sqrt{1}\right)+2\left(\sqrt{3}-\sqrt{2}\right)+...+2\left(\sqrt{101}-\sqrt{100}\right)\)
\(S>2\left(\sqrt{101}-1\right)>2\left(\sqrt{100}-1\right)=18\) (1)
\(S< 1+\frac{2}{\sqrt{1}+\sqrt{2}}+\frac{2}{\sqrt{2}+\sqrt{3}}+...+\frac{2}{\sqrt{99}+\sqrt{100}}\)
\(S< 1+2\left(\sqrt{2}-1\right)+2\left(\sqrt{3}-\sqrt{2}\right)+...+2\left(\sqrt{100}-\sqrt{99}\right)\)
\(S< 1+2\left(\sqrt{100}-1\right)=19\) (2)
(1); (2) \(\Rightarrow18< S< 19\)
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