Question

Chứng minh A < 2.

Answer 1

 \(A=\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+....+\frac{1}{50^2}\)<\(1+\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+\frac{1}{5.6}........+\frac{1}{49.50}\)

TA CO \(1+\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+\frac{1}{5.6}........+\frac{1}{49.50}\)

=1+\(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-.....+\frac{1}{49}-\frac{1}{50}\)

=1+\(\frac{25}{50}-\frac{1}{50}\)

=1+\(\frac{24}{50}\)

=1\(\frac{24}{50}\)

TA CO : 1\(\frac{24}{50}\)<2

MA A<1\(\frac{24}{50}\)

=>  A<2