Question

Bài 4

CMR:1/2^2+1/3^2+...+1/2013^2<1

Answer 1

\(\dfrac{1}{2^2}+\dfrac{1}{3^2}+....+\dfrac{1}{2013^2}\\ =\dfrac{1}{2.2}+\dfrac{1}{3.3}+...+\dfrac{1}{2013.2013}\\ < \dfrac{1}{1.2}+\dfrac{1}{2.3}+...+\dfrac{1}{2012.2013}\\ -1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+...+\dfrac{1}{2012}-\dfrac{1}{2013}\\ =1-\dfrac{1}{2013}< 1\)

Answer 2

\(\dfrac{1}{2^2}+\dfrac{1}{3^2}+..+\dfrac{1}{2013^2}< \dfrac{1}{1.2}+\dfrac{1}{2.3}+...+\dfrac{1}{2012.2013}=1-\dfrac{1}{2}+\dfrac{1}{2}+....+\dfrac{1}{2012}-\dfrac{1}{2013}\)

\(\Rightarrow dpcm\)