$\dfrac{1}{2}+\dfrac{1}{3\sqrt{2}}+......+\dfrac{1}{2007\sqrt{2006}}$
$= \dfrac{1}{2}+\dfrac{\sqrt{2}}{3.2}+......+\dfrac{ \sqrt{2006} }{2006.2007}$
$= \dfrac{1}{2}+\sqrt{2}.(\dfrac{1}{2} - \dfrac{1}{3})+......+\sqrt{2006}.(\dfrac{1}{2006} - \dfrac{1}{2007}).$
$= \dfrac{1}{2}+\sqrt{2}.(\dfrac{1}{\sqrt{2}} - \dfrac{1}{\sqrt{3}}).(\dfrac{1}{\sqrt{2}} + \dfrac{1}{\sqrt{3}})+......+\sqrt{2006}.(\dfrac{1}{\sqrt{2006}} - \dfrac{1}{\sqrt{2007}}).(\dfrac{1}{\sqrt{2006}} + \dfrac{1}{\sqrt{2007}}).$
$= \dfrac{1}{2}+(\dfrac{1}{\sqrt{2}} - \dfrac{1}{\sqrt{3}}).(\dfrac{1}{2} + \dfrac{\sqrt{2}}{\sqrt{3}})+......+(\dfrac{1}{ \sqrt{2006} } - \dfrac{1}{\sqrt{2007}}).(\dfrac{1}{2006} + \dfrac{\sqrt{2006}}{\sqrt{2007}}).$
$< \dfrac{1}{2}+2.(\dfrac{1}{\sqrt{2}} - \dfrac{1}{\sqrt{3}}) + ...+2.(\dfrac{1}{ \sqrt{2006} } - \dfrac{1}{\sqrt{2007}}).$
$=\dfrac{1}{2}+2.\dfrac{1}{\sqrt{2}}- 2.\dfrac{1}{\sqrt{2007}}.$
$<2$
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