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Chứng minh \(S< \dfrac{91}{330}\)
\(S=\left(\dfrac{1}{101}+\dfrac{1}{102}+.....+\dfrac{1}{110}\right)+\left(\dfrac{1}{111}+....+\dfrac{1}{120}\right)+\left(\dfrac{1}{121}+......+\dfrac{1}{130}\right)\)
\(S< \left(\dfrac{1}{100}+\dfrac{1}{100}......+\dfrac{1}{100}\right)+\left(\dfrac{1}{110}+....+\dfrac{1}{110}\right)+\left(\dfrac{1}{120}+....+\dfrac{1}{120}\right)\)
\(S< \dfrac{66+60+65}{660}\)
\(S< \dfrac{181}{660}< \dfrac{182}{660}\)
+ Hay \(S< \dfrac{91}{330}\left(1\right)\)
Chứng minh \(\dfrac{1}{4}< S\)
\(S>\left(\dfrac{1}{110}\right)+.....+\left(\dfrac{1}{110}\right)+\left(\dfrac{1}{120}\right)+.....+\left(\dfrac{1}{120}\right)+\left(\dfrac{1}{130}\right)+......+\left(\dfrac{1}{130}\right)\)
\(S>\dfrac{1}{110}.10+\dfrac{1}{120}.10+\dfrac{1}{130}.10=\dfrac{1}{11}+\dfrac{1}{12}+\dfrac{1}{13}\)
\(S>\dfrac{156+143+132}{1716}\)
+ Hay \(S>\dfrac{1}{4}\left(2\right)\)
Từ (1) và (2) \(\Rightarrow\dfrac{1}{4}< S< \dfrac{91}{330}\)
