Question

Cho A = 1/1² + 1/2² + 1/3²  +...+1/50². Chứng minh A < 2.

Answer 1

đặt B=1/1.2+1/2.3+1/3.4+...+1/49.50

=1/1.2+1/2.3+1/3.4+...+1/49.50

=1-1/2+1/2-1/3+...+1/49-50

=1-1/20<1 (1)

A =1/1*1+1/2.2+1/3.3+...+1/50*50<1-1/2+1/2-1/3+...+1/49-1/50 (2)

từ (1),(2)=>A<B<2

=>A<2

Answer 2

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

ta có :

\(\frac{1}{2^2}<\frac{1}{1.2}\)

\(\frac{1}{3^2}<\frac{1}{2.3}\)

...

\(\frac{1}{50^2}<\frac{1}{49.50}\)

-->\(\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{50^2}<\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{49.50}\)

-> \(\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{50^2}<\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{49}-\frac{1}{50}\)

-->\(\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{50^2}<\frac{49}{50}<1\)

Answer 3

-> 1+ \(\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{50^2}<1+1\)

=> A<2