11/2mu2+1/3mu2+1/4mu2+...+1/9mu2 cmr 2/5<s<8/9
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\(1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{100^2}< 2\)
\(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{100^2}< \frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{99.100}\)
\(=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{99}-\frac{1}{100}\)
\(=1-\frac{1}{100}\)
\(=\frac{99}{100}< 1\left(đpcm\right)\)
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1/2^2+1/3^2+1/4^2+....+1/2005^2
ta có vì:1/2^2<1/2; 1/3^2 <1/2.....;1/2005^2<1/2
suy ra 1/2^2+1/3^2+1/4^2+....+1/2005^2<1/2
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Ta thấy :
\(\frac{1}{2^2}< \frac{1}{1.2}\)
\(\frac{1}{3^2}< \frac{1}{2.3}\)
......
\(\frac{1}{100^2}< \frac{1}{99.100}\)
\(\Rightarrow\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{100^2}< \frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{99.100}\)
\(\Rightarrow S< 1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{99}-\frac{1}{100}\)
\(\Rightarrow S< 1-\frac{1}{100}\)
Mà \(1-\frac{1}{100}< 1\)nên \(S< 1\)
Ủng hộ mk nha !!! *_*
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\(\left[1-\frac{1}{2^2}\right]\left[1-\frac{1}{3^2}\right]\left[1-\frac{1}{4^2}\right]...\left[1-\frac{1}{10^2}\right]\)
\(=\left[1-\frac{1}{4}\right]\left[1-\frac{1}{9}\right]\left[1-\frac{1}{16}\right]...\left[1-\frac{1}{100}\right]\)
\(=\frac{3}{4}\cdot\frac{8}{9}\cdot\frac{15}{16}\cdot...\cdot\frac{99}{100}\)
Tự tính :v
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S = 22 + 42 + 62 + ... + 202
= (2.1)2 + (2.2)2 + (2.3)2 ... (2.10)2
= 22.12 + 22.22 + 22.32 + ... + 22.102
= 22 (12 + 22 + ... + 102 )
= 4 . 385
= 1540
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Đặt A =\(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{2015^2}\)
A < \(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{2014.2015}\)
A < \(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{2014}-\frac{1}{2015}\)
A < \(1-\frac{1}{2015}\)< \(1\)
=> A < 1 (đpcm)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(A=\dfrac{1}{2^2}+\dfrac{1}{3^2}+\dfrac{1}{4^2}+................+\dfrac{1}{2008^2}\)
Ta thấy :
\(\dfrac{1}{2^2}< \dfrac{1}{1.2}\)
\(\dfrac{1}{3^2}< \dfrac{1}{2.3}\)
...................
\(\dfrac{1}{2008^2}< \dfrac{1`}{2007.2008}\)
\(\Leftrightarrow A< \dfrac{1}{1.2}+\dfrac{1}{2.3}+............+\dfrac{1}{2007.2008}\)
\(\Leftrightarrow A< 1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+..........+\dfrac{1}{2007}-\dfrac{1}{2008}\)
\(\Leftrightarrow A< 1-\dfrac{1}{2008}< 1\)
\(\Leftrightarrow A< 1\rightarrowđpcm\)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(B=\dfrac{2^2}{1.3}.\dfrac{3^2}{2.4}.\dfrac{4^2}{3.5}...\dfrac{100^2}{99.101}=\dfrac{2.3.4...100}{1.2.3...99}.\dfrac{2.3.4..100}{3.4.5...101}=100.\dfrac{2}{101}=\dfrac{200}{101}\)