Question

Chứng minh \(\frac{1}{2^2}\)+ \(\frac{1}{3^2}\)+ \(\frac{1}{4^2}\)+ .... + \(\frac{1}{2009^2}\)< 1

Answer 1

\(Cm:\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{2009^2}< 1\)

Có : \(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{2009^2}< \frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+...+\frac{1}{2008\cdot2009}\)

\(\Leftrightarrow\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{2009^2}< \frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{2008}-\frac{1}{2009}\)

\(\Leftrightarrow\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{2009^2}< \frac{1}{1}-\frac{1}{2009}\)

\(\Rightarrow\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{2009^2}< \frac{2008}{2009}\left(1\right)\)

Mà \(\frac{2008}{2009}< 1\left(2\right)\)

\(\left(1\right),\left(2\right)\Rightarrow\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{2009^2}< \frac{2008}{2009}< 1\)

\(\Leftrightarrow\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{2009^2}< 1\left(đpcm\right)\)

Answer 2

\(\frac{1}{2^2}+\frac{1}{3^2}+....+\frac{1}{2009^2}< \frac{1}{1.2}+\frac{1}{2.3}+....+\frac{1}{2008.2009}\\ =\frac{1}{1}-\frac{1}{2009}< 1\left(\text{đ}pcm\right)\)