Question

Chứng minh rằng

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

(n thuộc N , n>2)

Answer 1

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

\(N=\frac{1}{2^2}.\left(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{n^2}\right)< \frac{1}{2^2}.\left(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+\frac{...1}{\left(n-1\right).n}\right)\)

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

\(N< \frac{1}{4}.\left(1-\frac{1}{n}\right)< \frac{1}{4}.1=\frac{1}{4}\)

=> \(N< \frac{1}{4}\)(đpcm)