Ta có:
Đặt A = $\frac{1}{2²}$+$\frac{1}{3²}$+$\frac{1}{4²}$+....+$\frac{1}{2009²}$
Ta có: $\frac{1}{2²}$<$\frac{1}{1.2}$
$\frac{1}{3²}$<$\frac{1}{2.3}$
$\frac{1}{4²}$<$\frac{1}{3.4}$
....
$\frac{1}{2009²}$<$\frac{1}{2008.2009}$
⇒ A <$\frac{1}{1.2}$+$\frac{1}{2.3}$+$\frac{1}{3.4}$+....+$\frac{1}{2008.2009}$
A < $\frac{1}{1}$- $\frac{1}{2}$ +$\frac{1}{2}$- $\frac{1}{3}$ +$\frac{1}{3}$- $\frac{1}{4}$+...+ +$\frac{1}{2008}$- $\frac{1}{2009}$
A < 1 - $\frac{1}{2009}$ <1 (đpcm)
+) "1.2+2.3+.....+2008.2009" bn hỏi tke thì mik chịu :vv