Question

Cmr:

C=1/4² + 1/6² + 1/8² +....+ 1/(2n)² < 1/4

D= 2!/3! + 2!/4! + 2!/5! +......+ 2!/n! < 1

Answer 1

Mình nghĩ gần 30 phút mới ra bài này ó; công nhận khó thật!!!

\(C=\frac{1}{4^2}+\frac{1}{6^2}+....+\frac{1}{\left(2n\right)^2}\\ =\frac{1}{2^2}\left(\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{n^2}\right)\\ < \frac{1}{4}\left(\frac{1}{1.2}+\frac{1}{2.3}+....+\frac{1}{\left(n-1\right)n}\right)\\ =\frac{1}{4}\left(\frac{1}{1}-\frac{1}{n}\right)< \frac{1}{4}\left(\text{đ}pcm\right)\)

\(D=\frac{2!}{3!}+\frac{2!}{4!}+....+\frac{2!}{n!}\\ =2!\left(\frac{1}{3!}+\frac{1}{4!}+....+\frac{1}{n!}\right)\\ < 2\left(\frac{1}{1.2.3}+\frac{1}{2.3.4}+....+\frac{1}{\left(n-2\right)\left(n-1\right)n}\right)=2\left(\frac{1}{2}\left(\frac{1}{1.2}-\frac{1}{\left(n-1\right)n}\right)\right)\\ =1\left(\frac{1}{2}-\frac{1}{\left(n-1\right)n}\right)< 1\left(\text{đ}pcm\right)\)

Chúc bạn học tốt !!!!!