[TEX]A=\frac{1}{2}-\frac{1}{3}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{98}-\frac{1}{99}[/TEX]
[TEX]\Rightarrow -A=\frac{-1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-...-\frac{1}{98}+\frac{1}{99}[/TEX]
[TEX]\Rightarrow -A=(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+...+\frac{1}{98}+\frac{1}{99})-2.(\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+...+\frac{1}{98})[/TEX]
[TEX]\Rightarrow -A=(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+...+\frac{1}{98}+\frac{1}{99})-(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{49})[/TEX]
[TEX]\Rightarrow -A= -1+\frac{1}{50}+\frac{1}{51}+..+\frac{1}{99}[/TEX]
Đặt: [TEX]B = \frac{1}{50}+\frac{1}{51}+..+\frac{1}{99}[/TEX]
\Rightarrow[TEX]B=(\frac{1}{50}+\frac{1}{51}+..+\frac{1}{59})+(\frac{1}{60}+\frac{1}{61}+..+\frac{1}{69})+(\frac{1}{70}+\frac{1}{71}+..+\frac{1}{79})+(\frac{1}{80}+\frac{1}{81}+..+\frac{1}{89})+(\frac{1}{90}+\frac{1}{91}+..+\frac{1}{99})[/TEX]
Do đó:
[TEX]10.\frac{1}{59}+10.\frac{1}{69}+10.\frac{1}{79}+10.\frac{1}{89}+10.\frac{1}{99}<B<10.\frac{1}{50}+10.\frac{1}{60}+10.\frac{1}{70}+10.\frac{1}{80}+10.\frac{1}{90}[/TEX]
\Rightarrow [TEX]10.\frac{1}{60}+10.\frac{1}{70}+10.\frac{1}{80}+10.\frac{1}{90}+10.\frac{1}{100}<B<10.\frac{1}{50}+10.\frac{1}{60}+10.\frac{1}{70}+10.\frac{1}{80}+10.\frac{1}{90}[/TEX]
\Rightarrow [TEX]1\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}<B<\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}[/TEX]
[TEX]=> 0,6456 < B < 0,7456[/TEX]
[TEX]=> \frac{3}{5} < B < \frac{4}{5}[/TEX]
[TEX]=> \frac{-2}{5} < -1+B < \frac{-1}{5}[/TEX]
[TEX]=> \frac{-2}{5} < -A < \frac{-1}{5}[/TEX]
[TEX]=> \frac{2}{5} > A > \frac{1}{5}[/TEX]
[TEX]\Rightarrow 0,4 >A>0,2 [/TEX] (dpcm)
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