\(A=\dfrac{1}{2^2}+\dfrac{1}{3^2}+\dfrac{1}{4^2}+................+\dfrac{1}{2008^2}\)

Ta thấy :

\(\dfrac{1}{2^2}< \dfrac{1}{1.2}\)

\(\dfrac{1}{3^2}< \dfrac{1}{2.3}\)

...................

\(\dfrac{1}{2008^2}< \dfrac{1`}{2007.2008}\)

\(\Leftrightarrow A< \dfrac{1}{1.2}+\dfrac{1}{2.3}+............+\dfrac{1}{2007.2008}\)

\(\Leftrightarrow A< 1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+..........+\dfrac{1}{2007}-\dfrac{1}{2008}\)

\(\Leftrightarrow A< 1-\dfrac{1}{2008}< 1\)

\(\Leftrightarrow A< 1\rightarrowđpcm\)

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 !!! *_*

\(1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{100^2}< 2\)

\(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{100^2}< \frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{99.100}\)

\(=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{99}-\frac{1}{100}\)

\(=1-\frac{1}{100}\)

\(=\frac{99}{100}< 1\left(đpcm\right)\)

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

\(\left[1-\frac{1}{2^2}\right]\left[1-\frac{1}{3^2}\right]\left[1-\frac{1}{4^2}\right]...\left[1-\frac{1}{10^2}\right]\)

\(=\left[1-\frac{1}{4}\right]\left[1-\frac{1}{9}\right]\left[1-\frac{1}{16}\right]...\left[1-\frac{1}{100}\right]\)

\(=\frac{3}{4}\cdot\frac{8}{9}\cdot\frac{15}{16}\cdot...\cdot\frac{99}{100}\)

Tự tính :v

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

= (1x2)^2 (2x2)^2 (3x2)^2 (4x2)^2 ..... (9x2)^2 (10x2)^2 
= 1^2 x 2^2 2^2 x 2^2 3^2 x 2^2 4^2 x 2^2 ..... 9^2 x 2^2 10^2 x 2^2 
= (1^2 2^2 3^2 4^2 ..... 9^2 10^2) x 2^2 
= 385 x 2^2 = 385 x 4 = 1540

\(B=\dfrac{2^2}{1.3}.\dfrac{3^2}{2.4}.\dfrac{4^2}{3.5}...\dfrac{100^2}{99.101}=\dfrac{2.3.4...100}{1.2.3...99}.\dfrac{2.3.4..100}{3.4.5...101}=100.\dfrac{2}{101}=\dfrac{200}{101}\)