ta có: \(\frac{1}{2^2}=\frac{1}{2.2}<\frac{1}{1.2};\frac{1}{3^2}=\frac{1}{3.3}<\frac{1}{2.3};...;\frac{1}{2010^2}=\frac{1}{2010.2010}<\frac{1}{2009.2010}\)
\(\Rightarrow N<\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{2009.2010}=\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+..+\frac{1}{2009}-\frac{1}{2010}=\frac{1}{1}-\frac{1}{2010}=\frac{2009}{2010}<1\)
=>N<1(đpcm)