CMR:[tex]P=\frac{1}{2^{2}}+\frac{1}{3^{2}}+\frac{1}{4^{2}}+.....+\frac{1}{100^{2}}< 1[/tex]
$ P = \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + ... + \frac{1}{100^2} < \frac{1}{1 . 2} + \frac{1}{2 . 3} + \frac{1}{3 . 4} + ... + \frac{1}{99 . 100} \\ S = \frac{1}{1 . 2} + \frac{1}{2 . 3} + \frac{1}{3 . 4} + ... + \frac{1}{99 . 100} \\ = 1 - \frac{1}{2} + \frac{1}{2} - \frac{1}{3} + \frac{1}{3} - \frac{1}{4} + ... + \frac{1}{99} - \frac{1}{100} \\ = 1 - \frac{1}{100} < 1 \\ P < S < 1 \\\Leftrightarrow P < 1 $