Ta có: $\frac{1}{2^2}$ < $\frac{1}{1.2}$ = $\frac{1}{1}$ : $\frac{1}{3^2}$ < $\frac{1}{2.3}$ = $\frac{1}{2}$ - $\frac{1}{3}$ : ... : $\frac{1}{100^2}$ < $\frac{1}{99.100}$ = $\frac{1}{99}$ - $\frac{1}{100}$
=> S < 5 ( 1 - $\frac{1}{100}$ < 5.1 = 5 => S<5 )
lại có : $\frac{1}{2^2}$ > $\frac{1}{2.3}$ = $\frac{1}{2}$ - $\frac{1}{3}$ : $\frac{1}{3^2}$ > $\frac{1}{3.4}$ = $\frac{1}{3}$ - $\frac{1}{4}$ : $\frac{1}{100^2}$ > $\frac{1}{100.101}$ = $\frac{1}{100}$ - $\frac{1}{101}$
=> S>5 ( $\frac{1}{2}$ - $\frac{1}{101}$ ) = 5. $\frac{101-2}{2.101}$ = $\frac{5.99}{2.101}$ . 2,45 => S > 2
vậy 2<S <5 => Đpcm
