Question

Cho A = 1/2² + 1/2² + 1/3² + ... + 1/50²

Chứng minh A < 2

Answer 1

\(A=\frac{1}{2^2}+\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{50^2}=\frac{1}{4}+\frac{1}{2.2}+\frac{1}{3.3}+...+\frac{1}{50.50}\)

\(<\frac{1}{4}+\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{49.50}=\frac{1}{4}+\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{49}-\frac{1}{50}\)

\(=\frac{1}{4}+\frac{1}{1}-\frac{1}{50}=\frac{123}{100}<2\)

Vậy A<2

Answer 2

Ta có:A=1/2^2+(1/2^2+1/3^2+......+1/50^2)

1/2^2<1/1.2

1/3^2<1/2.3

......

1/50^2<1/49.50

=>1/2^2+1/3^2+......+1/50^2<1/1.2+1/2.3+......+1/49.50=1/1-1/2+1/2-1/3+.......+1/49-1/50=1/1-1/50=49/50<1

=> A<1/2^2+1=5/4<8/4=2

 Vậy A<2( đpcm)

 

Answer 3

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