Question

[Nguyễn Linh]

cho A=1/2*2+1/3*2+...+1/2012*2+1/2013*2

so sánh A với 1

Answer 1

[Snow Princess]

\(A=\dfrac{1}{2^2}+\dfrac{1}{3^2}+...+\dfrac{1}{2012^2}+\dfrac{1}{2013^2}\)

\(A< \dfrac{1}{2.3}+\dfrac{1}{3.4}+...+\dfrac{1}{2012.2013}+\dfrac{1}{2013.2014}\)

\(A< \dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{2012}-\dfrac{1}{2013}+\dfrac{1}{2013}-\dfrac{1}{2014}\)

\(A< \dfrac{1}{2}-\dfrac{1}{2014}\)

\(A< \dfrac{1007}{2014}-\dfrac{1}{2014}\)

\(A< \dfrac{1006}{2014}< 1\)

\(\Rightarrow A< 1\)