Question

A = 1/2² +1/3²+1/4²+.....+1/2019² . So Sánh A với 3/4

Giúp vs mn mai mik đi hc òi

Answer 1

\(2^2>1.3\); \(3^2>2.4\) ; \(n^2>\left(n-1\right)\left(n+1\right)\)

\(\Rightarrow A< \frac{1}{1.3}+\frac{1}{2.4}+\frac{1}{3.5}+...+\frac{1}{2018.2020}\)

\(A< \frac{1}{2}\left(1-\frac{1}{3}+\frac{1}{2}-\frac{1}{4}+\frac{1}{3}-\frac{1}{5}+...+\frac{1}{2018}-\frac{1}{2020}\right)\)

\(A< \frac{1}{2}\left(1+\frac{1}{2}-\frac{1}{2020}\right)< \frac{1}{2}\left(1+\frac{1}{2}\right)=\frac{3}{4}\)

\(\Rightarrow A< \frac{3}{4}\)