Question

Bài 4

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

Answer 1

\(\dfrac{1}{2^2}< \dfrac{1}{1.2}\)

\(\dfrac{1}{3^2}< \dfrac{1}{2.3}\)

....................

\(\dfrac{1}{2013^2}< \dfrac{1}{2012.2013}\)

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

\(\Leftrightarrow\dfrac{1}{2^2}+\dfrac{1}{3^2}+......+\dfrac{1}{2013^2}< 1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+......+\dfrac{1}{2012}-\dfrac{1}{2013}\)

\(\Leftrightarrow\dfrac{1}{2^2}+\dfrac{1}{3^2}+......+\dfrac{1}{2013^2}< 1-\dfrac{1}{2013}\)

\(\Leftrightarrow\dfrac{1}{2^2}+\dfrac{1}{3^2}+.....+\dfrac{1}{2013^2}< 1\left(đpcm\right)\)