Question

cho A=1/1^2+1/2^2+1/3^2+1/4^2+...+1/50^2

chứng minh A<2

Answer 1

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

\(A=1+\left(\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{50^2}\right)=1+B\)( Gọi biểu thức trong ngoặc là B)

Ta xét B

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

B<\(\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{49.50}\)

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

B<\(1-\frac{1}{50}<1\)

Vậy B<1

=>A=1+B < 1+1=2

Vậy A<2