Question

CMR : S < 1/4 với S = 1/4^2 +1/6^2 + 1/8^2 + ... 1/(2n)^2

 

Answer 1

\(S=\frac{1}{4^2}+\frac{1}{6^2}+\frac{1}{8^2}+...+\frac{1}{\left(2n\right)^2}=\frac{1}{4}\left(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{n^2}\right)\)

Ta có :\(\frac{1}{2^2}< \frac{1}{1.2};\frac{1}{3^2}< \frac{1}{2.3};\frac{1}{4^2}< \frac{1}{3.4};....;\frac{1}{n^2}< \frac{1}{\left(n-1\right)n}\)

\(\Rightarrow S< \frac{1}{4}\left(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+....+\frac{1}{\left(n-1\right)n}\right)\)

\(\Leftrightarrow S< \frac{1}{4}\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+....+\frac{1}{n-1}-\frac{1}{n}\right)\)

\(\Rightarrow S< \frac{1}{4}\left(1-\frac{1}{n}\right)< \frac{1}{4}\) (đpcm)