Question

cho A= 1/2^2 + 1/3^2 + ... + 1/2017^2. Chứng minh A<3/4

Answer 1

Ta có : \(\frac{1}{2^2}=\frac{1}{4}\)

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

\(\frac{1}{4^2}< \frac{1}{3.4}\)

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

\(\frac{1}{2017^2}< \frac{1}{2016.2017}\)

\(\Rightarrow A< \frac{1}{4}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{2016.2017}\)

\(\Rightarrow A< \frac{1}{4}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{2016}-\frac{1}{2017}\)

\(\Rightarrow A=\frac{1}{4}+\frac{1}{2}-\frac{1}{2017}\)

\(A=\frac{3}{4}-\frac{1}{2017}\left(đpcm\right)\) . Vậy A < \(\frac{3}{4}\)