Question

Cho A=\(1-\frac{1}{2^2}-\frac{1}{3^2}-....-\frac{1}{2010^2}\)

CMR : A > \(\frac{1}{2010}\)

GIÚP MIK VS!!

Answer 1

\(A=1-\frac{1}{2^2}-...-\frac{1}{2010^2}\)

\(=1-\left(\frac{1}{2^2}+...+\frac{1}{2010^2}\right)\)

Đặt \(B=\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{2009.2010}\)

Ta có: \(A=1-\left(\frac{1}{2^2}+...+\frac{1}{2010^2}\right)\)\(>\)\(B=1-\left(\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{2009.2010}\right)\)

\(=1-\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{2009}-\frac{1}{2010}\right)\)

\(=1-\left(1-\frac{1}{2010}\right)=1-1+\frac{1}{2010}=\frac{1}{2010}\)