• Chứng minh $S < \frac{91}{330}$:
$S = (\frac{1}{101} + \frac{1}{102} + ... + \frac{1}{110}) + (\frac{1}{111} + ... + \frac{1}{120}) + (\frac{1}{121} + ... + \frac{1}{130})$.
$S < (\frac{1}{100} + \frac{1}{100} + ... + \frac{1}{100}) + (\frac{1}{110} + ... + \frac{1}{110}) + (\frac{1}{120} + ... + \frac{1}{120})$.
$S < \frac{1}{100}.10 + \frac{1}{110}.10 + \frac{1}{120}.10 = \frac{1}{10} + \frac{1}{11} + \frac{1}{12}$.
$S < \frac{66 + 60 + 55}{660}$.
$S < \frac{181}{660} < \frac{182}{660}$.
+ Hay: $S < \frac{91}{330}$. $(1)$
• Chứng minh $\frac{1}{4} < S$:
$S > (\frac{1}{110}) + ... + (\frac{1}{110}) + (\frac{1}{120} + ... + (\frac{1}{120}) + (\frac{1}{130} + ... + \frac {1}{130})$.
$S > \frac{1}{110}.10 + \frac{1}{120}.10 + \frac{1}{130}.10 = \frac{1}{11} + \frac{1}{12} + \frac{1}{13}$.
$S > \frac{156 + 143 + 132}{1716}$.
$S > \frac{431}{1716} > \frac{429}{1716}$.
+ Hay: $S > \frac{1}{4}$. $(2)$
+ Từ $(1)$ và $(2)$ $⇒ \frac{1}{4} < S < \frac{91}{330}$.
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