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![](https://rs.olm.vn/images/avt/0.png?1311)
\(A=\frac{1}{2}\times\frac{3}{4}......\frac{9999}{10000}\)
Đặt : \(B=\frac{2}{3}\times\frac{4}{5}\times\frac{6}{7}.......\frac{10000}{10001}\)
Vì \(\frac{1}{2}< \frac{2}{3};\frac{3}{4}< \frac{4}{5};.....\frac{9999}{10000}< \frac{10000}{10001}\)
Nên A<B mà A>0; B>0
\(\Rightarrow A^2< A\times B=\left(\frac{1}{2}\times\frac{3}{4}\times\frac{5}{6}.....\frac{9999}{10000}\right)\times\left(\frac{2}{3}\times\frac{4}{5}\times\frac{6}{7}......\frac{10000}{10001}\right)\)\(=\frac{1}{2}\times\frac{2}{3}\times\frac{4}{5}......\frac{9999}{10000}\times\frac{10000}{10001}\)\(=\frac{1}{10001}< \frac{1}{10000}=\frac{1}{100^2}=0.01^2\)\(\Rightarrow A^2< 0.01^2\)hay A < 0.01
![](https://rs.olm.vn/images/avt/0.png?1311)
![](https://rs.olm.vn/images/avt/0.png?1311)
đặt C=1/2.3/4.5/6....9999/10000
đặt B=2/3.4/5.6/7....10000/10001
C.B=1/2.2/3.3/4.4/5.5/6.6/7....9999/10000.10000/10001
C.B=1/10001,đặt C.B=A
vì 1/10001<0,01=> A<0,01
![](https://rs.olm.vn/images/avt/0.png?1311)
Ta có: \(0,01=\frac{1}{100}\)
Mà \(A=\frac{1}{2}.\frac{3}{4}.\frac{5}{6}.....\frac{9999}{10000}\)
Ta thấy: \(\frac{1}{100}=\frac{100}{10000}\)
Vì \(\frac{9999}{10000}>\frac{100}{10000}hay\frac{9999}{10000}>\frac{1}{100}\)
Nên \(\frac{1}{2}.\frac{3}{4}.\frac{5}{6}.....\frac{9999}{10000}>\frac{1}{100}hay\frac{1}{2}.\frac{3}{4}.\frac{5}{6}.....\frac{9999}{10000}>0,01\)
Vậy \(A>0,01\)
Ta có: \(0,01=\frac{1}{100}\)
Đặt \(B=\frac{2}{3}.\frac{4}{5}.\frac{6}{7}....\frac{10000}{10001}\)
Xét \(AB=\frac{1}{2}.\frac{3}{4}.\frac{5}{6}....\frac{9999}{10000}.\frac{2}{3}.\frac{4}{5}.\frac{6}{7}....\frac{10000}{10001}\)
\(\Leftrightarrow\)\(AB=\frac{1.2.3.4.5.6.....9999.10000}{2.3.4.5.6.7.....10000.10001}\)
\(\Leftrightarrow\)\(AB=\frac{1}{10001}\)
Vì A < B
\(\Rightarrow\)A2 < AB
\(\Rightarrow A^2< \frac{1}{10001}< \frac{1}{10000}\)
\(\Rightarrow A< \frac{1}{100}hayA< 0,01\)
Vậy A < 0,01
![](https://rs.olm.vn/images/avt/0.png?1311)
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