Question

Cho \(S=\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{50^2}\)

50 mũ 2 nhé

Chứng minh rằng S<\(\frac{3}{4}\)

Answer 1

\(S=\frac{1}{4}+\left(\frac{1}{3^2}+\frac{1}{4^2}+..+\frac{1}{50^2}\right)\)

Xét \(A=\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{50^2}\)

\(A< \frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{49.50}\)

\(A< \frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{49}-\frac{1}{50}\)

\(A< \frac{1}{2}-\frac{1}{50}< \frac{1}{2}\)

\(=>A< \frac{1}{2}\)

=>\(S=\frac{1}{4}+A< \frac{1}{4}+\frac{1}{2}=\frac{3}{4}\)

vậy S<3/4