Ta có:
\(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+.....+\frac{1}{2014^2}\)
\(< \frac{1}{4}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+.....+\frac{1}{2013\cdot2014}\)
\(=\frac{1}{4}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+.....+\frac{1}{2013}-\frac{1}{2014}\)
\(=\frac{1}{4}+\frac{1}{2}-\frac{1}{2014}\)
\(=\frac{3}{4}-\frac{1}{2014}\)
\(< \frac{3}{4}\)
\(\text{Cho }A=\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{2014^2}\text{ Chứng tỏ }A< \frac{3}{4}\)
Cho \(A=\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{2014^2}\) . Chứng tỏ \(A< \frac{3}{4}\)
a)\(\frac{7}{x}<\frac{x}{4}<\frac{10}{x}\)
b) Cho A=\(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{9^2}\). Chứng tỏ: \(\frac{8}{9}>A>\frac{2}{5}\)
Chứng tỏ:
a) \(\frac{2013}{2014}+\frac{2014}{2015}+\frac{2015}{2013}>3\)
b) \(\left(1+\frac{1}{2}\right)\left(1+\frac{1}{2^2}\right)\left(1+\frac{1}{2^3}\right)\left(1+\frac{1}{2^4}\right)....\left(1+\frac{1}{2^{50}}\right)< 3\)
c) \(C=\frac{1}{2}.\frac{3}{4}.\frac{5}{6}.....\frac{9999}{10000}< \frac{1}{100}\)
d) \(\frac{1}{2}-\frac{1}{2^2}+.............+\frac{1}{2^{99}}-\frac{1}{2^{100}}< \frac{1}{3}\)
Chứng tỏ:
a) \(\frac{2013}{2014}+\frac{2014}{2015}+\frac{2015}{2013}>3\)
b) \(\left(1+\frac{1}{2}\right)\left(1+\frac{1}{2^2}\right)\left(1+\frac{1}{2^3}\right)\left(1+\frac{1}{2^4}\right)...\left(1+\frac{2}{50}\right)< 3\)
c) \(C=\frac{1}{2}.\frac{3}{4}.\frac{5}{6}.....\frac{9999}{10000}< \frac{1}{100}\)
d) \(\frac{1}{2}-\frac{1}{2^2}+.........+\frac{1}{2^{99}}-\frac{1}{2^{100}}< \frac{1}{3}\)
\(A=\left(\frac{1}{2^2}-1\right)\left(\frac{1}{3^2}-1\right)\left(\frac{1}{4^2}-1\right).....\left(\frac{1}{2013^2}-1\right)\left(\frac{1}{2014^2}-1\right)\)
CHỨNG TỎ : A<-1/2
Chứng minh: \(A=\frac{1}{3}-\frac{2}{3^2}+\frac{3}{3^3}-\frac{4}{3^4}+...-\frac{2014}{3^{2014}}<\frac{1}{5}\)
cho S=\(\frac{1}{4^1}+\frac{2}{4^2}+\frac{3}{4^3}+...+\frac{2014}{4^{2014}}\)
chứng minh S<\(\frac{1}{2}\)
Cho A= \(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{100^2}\)Chứng tỏ A < \(\frac{3}{4}\)
