Ta có:
\(\sqrt{2\sqrt{3\sqrt{4...\sqrt{2000}}}}\)
\(< \sqrt{2\sqrt{3\sqrt{4...\sqrt{2000.2002}}}}\)
\(=\sqrt{2\sqrt{3\sqrt{4...\sqrt{1999\sqrt{2001^2-1}}}}}\)
\(< \sqrt{2\sqrt{3\sqrt{4...\sqrt{1999.2001}}}}\)
\(........................................\)
\(< \sqrt{2.4}=\sqrt{8}< 3\)
Ta có:
√2√3√4...√2000
<√2√3√4...√2000.2002
=√2√3√4...√1999√20012−1
<√2√3√4...√1999.2001
........................................
<√2.4=√8<3