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B=\(\left(1-\dfrac{1}{2}\right).\left(1-\dfrac{1}{3}\right).\left(1-\dfrac{1}{4}\right)...\left(1-\dfrac{1}{20}\right)\)
\(=\dfrac{1}{2}.\dfrac{2}{3}.\dfrac{3}{4}...\dfrac{19}{20}\)
\(=\dfrac{1.2.3....19}{2.3.4.....20}\)
\(=\dfrac{1.2.3....19:\left(2.3.....19\right)}{2.3.4.....20:\left(2.3.4.....19\right)}\)
\(=\dfrac{1}{20}\)
A<1-1/2+1/2-1/3+...+1/8-1/9=1-1/9=8/9 A>1/2-1/3+1/3-1/4+...+1/9-1/10=1/2-1/10=2/5 =>2/5<A<8/9
\(\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{100^2}< \frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{99.100}\)
\(\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{100^2}< \frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{99}-\frac{1}{100}\)
\(\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{100^2}< \frac{1}{2}-\frac{1}{100}\)
\(\Rightarrow\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{100^2}< \frac{1}{2}\)
Gọi A = \(\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{100^2}\)
=> A = \(\frac{1}{3.3}+\frac{1}{4.4}+...+\frac{1}{100.100}\)
A < \(\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{99.100}\)
A < \(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{99}-\frac{1}{100}\)
A < \(\frac{1}{2}-\frac{1}{100}\)
A < \(\frac{49}{100}< \frac{50}{100}=\frac{1}{2}\)
=> A < \(\frac{1}{2}\)
<=> \(\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{100^2}< \frac{1}{2}\)
\(-\frac{1}{2}+\frac{1}{3}+\left(-\frac{1}{4}\right)+\left(-\frac{2}{8}\right)+\frac{4}{18}+\frac{4}{9}\)
= \(0\)
A: x thuộc -5 ; 7
B: tập hợp rỗng
C: x thuộc -3 ; 5
D: x thuộc -8; 4
\(A=1+\dfrac{1}{2}+\dfrac{1}{2^2}+...+\dfrac{1}{2^{2012}}\)
\(2A=2+1+\dfrac{1}{2}+...+\dfrac{1}{2^{2011}}\)
\(2A-A=2+1+\dfrac{1}{2}+...+\dfrac{1}{2^{2011}}-1-\dfrac{1}{2}-\dfrac{1}{2^2}-...-\dfrac{1}{2^{2012}}\)
\(A=2-\dfrac{1}{2^{2012}}\)