Ta có : 1/2^2<1/1.2
1/3^2 < 1/2.3
1/4^2<1/3.4
................
.............
1/2010^2<1/2009.2010
=> 1/2^2+1/3^2+1/4^2+1/5^2+.....+1/2010^2 < 1/1.2+1/2.3+1/3.4+....+1/2009.2010
=> 1/2^2+1/3^2+1/4^2+1/5^2+.....+1/2010^2 < 1-1/2009
=> 1/2^2+1/3^2+1/4^2+1/5^2+.....+1/2010^2 < 2008/2009 < 1
Vậy 1/2^2+1/3^2+1/4^2+1/5^2+.....+1/2010 < 1
\(\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{2010^2}<\frac{1}{1.2}+\frac{1}{2.3}+....+\frac{1}{2009.2010}\)
\(=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+....+\frac{1}{2009}-\frac{1}{2010}=1-\frac{1}{2010}=\frac{2009}{2010}<1\)
