Question

1/2²⁺ 1/3²⁺1/4²+.......+1/2011²+1/2012² < 1

Answer 1

Ta có:

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

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

\(..............\)

\(\frac{1}{2012^2}< \frac{1}{2011.2012}\)

\(\Rightarrow A=\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{2012^2}< \frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{2011.2012}\)

\(\Rightarrow A=\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{2012^2}< 1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{2011}-\frac{1}{2012}\)

\(\Rightarrow A< 1-\frac{1}{2012}< 1\)

\(\)Vậy \(A< 1\left(đpcm\right)\)

Answer 2

1/22+ 1/32+1/42+.......+1/20112+1/20122 < 1